2.1. Description of continuous-time embedding

In the continuous-time approach, we begin with a population consisting of a single vertex 0 with weight $W_0$ sampled from $\mu$ and an associated exponential clock with parameter $f(0,W_0)$ . Then recursively, when the ith birth event occurs in the population, with the ringing of an exponential clock associated to vertex j, the following occurs:

1. (i) Vertex j produces offspring $\ell (i-1) + 1, \ldots, \ell i$ with independent weights $W_{\ell (i-1) +1}, \ldots, W_{\ell i}$ sampled from $\mu$ and exponential clocks with parameters $f(0, W_{\ell (i-1) +1}), \ldots, f(0, W_{\ell i})$ .

2. (ii) Suppose the number of offspring of j before the birth event was m, so that its out-degree in the family tree is m. Then the exponential random variable associated with j is updated to have rate $f(m/\ell + 1, {W_{j}})$ . If $f(m/\ell + 1, {W_{j}}) = 0$ , then j ceases to produce offspring and we say j has died.

Now, if we let ${\mathcal{Z}}_{i-1}$ denote the sum of the rates of the exponential clocks in the population when the population has size $i-1$ , the probability that the clock associated with j is the first to ring is $f(m/\ell, W_j)/{\mathcal{Z}}_{i-1}$ . Hence, the family tree of the continuous-time model at the times of the birth events $(\sigma_{i})_{i \geq 0}$ has the same distribution as the associated ${\left({\mu, f, \ell}\right)}$ -RIF tree. The continuous-time branching process is actually a Crump–Mode–Jagers branching process, which we will describe in more depth in Section 2.2.

To describe the evolution of the degree of a vertex in the continuous-time model, we define the pure birth process with underlying probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and state space $\ell \mathbb{N}$ as follows. First sample a weight W and set $Y(0) = 0$ . Let $\mathbb{P}_{w}$ denote the probability measure associated with the process when the weight sampled is w. Then define the birth rates of Y so that

(9) \begin{equation} {{\mathbb{P}_{w} \!\left({Y(t + h) = (k+1)\ell \; | \; Y(t) = k\ell}\right)}} = f(k, w) h + o(h).\end{equation}

In other words, the time taken to jump from $k\ell$ to $(k + 1)\ell$ is exponentially distributed with parameter f(k, w).

Let $\rho$ denote the point process corresponding to the jumps in Y, and denote by ${{\mathbb{E}_{w} \!\left[{\rho({\cdot})} \right]}}$ the intensity measure when the weight $W = w$ . Also, denote by $\hat{\rho}_w$ the Laplace–Stieltjes transform, i.e.,

\begin{equation*}\hat{\rho}_w(\lambda) \,:\!=\, \int_{0}^{\infty}e^{-\lambda t} {{\mathbb{E}_{w} \!\left[{\rho({\text{d}} t)} \right]}}.\end{equation*}

Note that, by Fubini’s theorem, we have

(10) \begin{align} \hat{\rho}_w(\lambda) & = \int_{0}^{\infty} \left(\int_{t}^{\infty} \lambda e^{- \lambda s} {\text{d}} s \right)\, {{\mathbb{E}_{w} \!\left[{\rho({\text{d}} t)} \right]}} & =\int_{0}^{\infty} \lambda e^{-\lambda s} \left(\int_{0}^{s} {{\mathbb{E}_{w} \!\left[{\rho({\text{d}} t)} \right]}}\right) {\text{d}} s\\ & = \int_{0}^{\infty} \lambda e^{-\lambda s} {{\mathbb{E}_{w} \!\left[{Y(s)} \right]}}{\text{d}} s. \nonumber \end{align}

Moreover, if we write $\tau_{k}$ for the time of the kth jump in Y, we have $\rho = \sum_{k=0}^{\infty} \ell \delta_{\tau_{k}}.$ Note that if the weight of Y is w, then $\tau_{k}$ is distributed as a sum of independent exponentially distributed random variables with rates $f(0,w), f(1,w), \ldots, f(k-1,w)$ , where we follow the convention that an exponentially distributed random variable with rate 0 is $\infty$ . Thus, we have that

(11) \begin{equation} \hat{\rho}_w(\lambda) = \ell \sum_{n=1}^{\infty} {{\mathbb{E}_{w} \!\left[{e^{-\lambda \tau_{n}}} \right]}} =\ell\sum_{n=1}^{\infty} \prod_{i=0}^{n-1} \frac{f(i,w)}{f(i,w) + \lambda},\end{equation}

where in the last equality we have used the facts that a Laplace–Stieltjes transform of a convolution of measures is the product of Laplace–Stieltjes transforms, and the Laplace–Stieltjes transform $\hat{X}(\lambda)$ of an exponentially distributed random variable with parameter s is $\int_{0}^{\infty} e^{-\lambda t} s e^{- s t} {\text{d}} t = \frac{s}{s+\lambda}$ . Therefore, we see that ${{\mathbb{E} \!\left[{\hat{\rho}_{W}(\lambda)} \right]}} = m(\lambda, \mathbb{R}_{+})$ as defined in (8), and Condition C1 implies that there exists some $\lambda > 0$ such that $1 < {{\mathbb{E}\!\left[{\hat{\rho}_{W}(\lambda)} \right]}} < \infty$ . In addition, the Malthusian parameter $\alpha$ appearing in Condition C1 is the unique positive real number such that

(12) \begin{equation} {{\mathbb{E}\!\left[{\hat{\rho}_{W}(\alpha)} \right]}} = m(\alpha, \mathbb{R}_{+}) = \ell \cdot {{\mathbb{E}\!\left[{\sum_{n=1}^{\infty} \prod_{i=0}^{n-1} \frac{f(i,W)}{f(i,W) + \alpha}} \right]}} = 1.\end{equation}

Our first result is the following.

Theorem 2.1. (Convergence of the degree distribution under C1.) Let ${\mathcal{T}}$ be a ${{\left({\mu, f, \ell}\right)}}$ -RIF tree satisfying C1 with Malthusian parameter $\alpha$ . Then, for any measurable set $B \subseteq \mathbb{R}_+$ , with $N_{k}(t, B)$ as defined in (1) and $p^{\alpha}_{k}(B)$ as defined in (3), we have

\begin{equation*}\frac{N_{k}(t, B)}{\ell t} \xrightarrow{t \to \infty} p^{\alpha}_{k}(B),\end{equation*}

almost surely.

The limiting formula for Theorem 2.1 has appeared in a number of contexts, and generalises many known results. Under Condition C1 this result was proved by Rudas, Tóth and Valkó [Reference Rudas, Tóth and Valkó33] in the case that W is constant and $\ell = 1$ . The cases $f(i,W) = W(i+1)$ and $f(i,W) = i+1+W$ with $\ell = 1$ correspond respectively to the preferential attachment models with multiplicative and additive fitness mentioned in the introduction. In the multiplicative model, the result was first proved in [Reference Borgs, Chayes, Daskalakis and Roch8] and later in [Reference Bhamidi3]. In [Reference Bhamidi3], Bhamidi also first proved the result for the case $f(i,W) = i + 1 + W$ . These models are examples of the generalised preferential attachment tree with fitness, which we study in more depth in Section 3. Finally, the case $f(i,W) = W$ , $\ell = 1$ corresponds to a model of weighted random recursive trees (see Example 2.4.2). We postpone the proof of Theorem 2.1 to the end of Section 2.2.

Remark 2.2. Note that $Y(t) < \infty$ for all $t \geq 0$ almost surely if $\tau_{\infty} \,:\!=\, \lim_{k \to \infty} \tau_{k} = \infty$ almost surely. The latter is satisfied if there exists $\lambda > 0 $ such that for almost all w

\begin{equation*}{{\mathbb{E}_{w} \!\left[{e^{-\lambda \tau_\infty}} \right]}} = \lim_{n \to \infty} {{\mathbb{E}_{w} \!\left[{e^{-\lambda \tau_n}} \right]}} = \lim_{n \to \infty}\prod_{i=0}^{n} \frac{f(i, w)}{f(i,w) + \lambda} = 0,\end{equation*}

which is implied by C1. In the literature concerning pure birth Markov chains, this property is known as non-explosivity.

Now, recall the definitions of $\Xi(t, \cdot)$ from (2) and $m(\alpha, \cdot)$ from (6). In the case that $m(\alpha, \cdot)$ is a probability distribution, the almost sure convergence of $N_{k}(t,B)/\ell n$ to $p^{\alpha}_{k}(B)$ for any measurable set B is enough to imply that for any measurable set B the quantity $\Xi(t,B)$ converges almost surely to $m(\alpha, B)$ . Note that this condition is weaker than directly assuming C1. In particular, we have the following.

Theorem 2.2. Assume ${\mathcal{T}}$ is a ${\left({\mu,f, \ell}\right)}$ -RIF tree with limiting degree distribution of the form $(p^{\alpha}_{k}({\cdot}))_{k \in \mathbb{N}_{0}}$ and such that $m(\alpha, \mathbb{R}_{+}) = 1$ . Then for any measurable set B we have

\begin{equation*}\frac{\Xi(t,B)}{\ell t} \xrightarrow{t \to \infty} m(\alpha, B),\end{equation*}

almost surely.

To prove this theorem, we will apply the following elementary bound: for any two sequences $(a_n)_{n \in \mathbb{N}}$ , $(b_{n})_{n \in \mathbb{N}}$ such that either $\liminf_{n \to \infty} a_n > -\infty$ or $\limsup_{n \to \infty} b_n < \infty$ , we have

(13) \begin{equation} \liminf_{n \to \infty}(a_n + b_n) \leq \liminf_{n \to \infty} a_n + \limsup_{n \to \infty} b_n \leq \limsup_{n \to \infty}(a_n + b_n).\end{equation}

Proof of Theorem 2.2. Recall that, by (4), for each t, we have $\Xi(t,B) = \sum_{k=1}^{t} k \ell N_{k}(t,B)$ . Also note that

\begin{align*}\sum_{k=0}^{\infty} k \ell p^{\alpha}_{k}(B)& = \ell \cdot {{\mathbb{E}\!\left[{\left(\sum_{k=1}^{\infty} \frac{k\alpha}{f(k,W) + \alpha} \prod_{i=0}^{k-1} \frac{f(i,W)}{f(i,W) + \alpha} \right) \mathbf{1}_{B}(W)} \right]}} \\ & = \ell \cdot {{\mathbb{E}\!\left[{\left(\sum_{k=1}^{\infty} k \cdot \left(1 - \frac{f(k,W)}{f(k,W) + \alpha}\right) \prod_{i=0}^{k-1} \frac{f(i,W)}{f(i,W) + \alpha} \right) \mathbf{1}_{B}(W)} \right]}}\\ & = \ell \cdot {{\mathbb{E}\!\left[{\sum_{k=1}^{\infty} \left(k\prod_{i=0}^{k-1} \frac{f(i,W)}{f(i,W) + \alpha} - k\prod_{i=0}^{k} \frac{f(i,W)}{f(i,W) + \alpha}\right)\mathbf{1}_{B}(W)} \right]}} \\ & =\ell \cdot {{\mathbb{E}\!\left[{\left(\sum_{k=1}^{\infty} \prod_{i=0}^{k-1} \frac{f(i,W)}{f(i,W) + \alpha}\right) \mathbf{1}_{B}(W)} \right]}} = m(\alpha, B),\end{align*}

where the second-to-last equality follows from the telescoping nature of the sum inside the expectation. Thus, by Fatou’s lemma, almost surely we have

(14) \begin{equation} m(\alpha, B) = \sum_{k=0}^{\infty} k \ell p^{\alpha}_{k}(B) = \sum_{k=0}^{\infty} k \ell \liminf_{t \to \infty} \frac{N_{k}(t,B)}{\ell t} \leq \liminf_{t \to \infty} \frac{\Xi(t,B)}{\ell t};\end{equation}

and likewise, almost surely, $\liminf_{t \to \infty} \frac{\Xi(t,B^c)}{\ell t} \geq m(\alpha, B^c)$ . Now, since we add $\ell$ edges at every time-step, $\Xi(t,\mathbb{R}_{+}) = \ell t$ . Thus, by (13),

\begin{align*}1 = \liminf_{t \to \infty} \left(\frac{\Xi(t, B)}{\ell t} + \frac{\Xi(t,B^{c})}{\ell t}\right)& \leq \liminf_{t \to \infty} \frac{\Xi(t, B^c)}{\ell t} + \limsup_{t \to \infty} \frac{\Xi(t,B)}{\ell t} \\ & \leq \limsup_{t \to \infty} \left(\frac{\Xi(t, B)}{\ell t} + \frac{\Xi(t,B^{c})}{\ell t}\right) = 1.\end{align*}

But $m(\alpha, \cdot)$ is a probability measure; this is only possible if

(15) \begin{equation} \liminf_{t \to \infty} \frac{\Xi(t, B^c)}{\ell t} = m(\alpha, B^{c}) \quad \text{ and } \quad \limsup_{t \to \infty} \frac{\Xi(t,B)}{\ell t} = m(\alpha, B) \quad \text{ almost surely.}\end{equation}

Combining (14) and (15) completes the proof.

2.2. Crump–Mode–Jagers branching processes

In the continuous-time setting, it is convenient not only to identify individuals of the branching process according to the order in which they were born, but also to record their lineage, in such a way that the labelling encodes the structure of the tree. Therefore we also identify individuals of the branching process with elements of the infinite Ulam–Harris tree $\mathcal{U} \,:\!=\, \bigcup_{n \geq 0} \mathbb{N}^{n}$ , where $\mathbb{N}^{0} = \varnothing$ is the root. In this case, an individual $u = u_1u_2\ldots u_k$ is to be interpreted recursively as the $u_k$ th child of $u_1 \ldots u_{k-1}$ . For example, $1, 2, \ldots$ represent the offspring of $\varnothing$ .

In Crump–Mode–Jagers (CMJ) branching processes, individuals $u \in \mathcal{U}$ are equipped with independent copies of a random point process $\xi$ on $\mathbb{R}_+$ . The point process $\xi$ associates birth times to the offspring of a given individual, and we also may assume that $\xi$ has some dependence on a random weight W associated with that individual. The process, together with birth times, may be regarded as a random variable in the probability space $(\Omega, \Sigma, \mathbb{P}) = \prod_{x \in \mathcal{U}} (\Omega_{x}, \Sigma_{x}, \mathbb{P}_{x})$ , where each $(\Omega_{x}, \Sigma_{x}, \mathcal{P}_{x})$ is a probability space with $(\xi_{x}, W_x)$ having the same distribution as $(\xi,W)$ . We denote by $(\sigma_{i}^{x})_{i \in \mathbb{N}}$ points ordered in the point process $\xi_{x}$ and, for brevity, assume that $\xi(\{0\}) = 0$ . We also drop the superscript when referring to the point process associated to $\varnothing$ , so that $\sigma_{i} \,:\!=\, \sigma^{\varnothing}_i$ . Now, we set $\sigma_{\varnothing} \,:\!=\, 0$ and recursively, for $x \in \mathcal{U}$ , $\sigma_{xi} \,:\!=\, \sigma_{x} + \sigma^{x}_{i}$ . Finally, we set $\mathbb{T}_{t} = \{x \in \mathcal{U}\,:\, \sigma_{x} \leq t\}$ and note that for each $t \geq 0$ , $\mathbb{T}_{t}$ may be identified with the family tree of the process in the natural way. Informally, $\mathbb{T}_{t}$ can be described as follows: at time zero, there is one vertex $\varnothing$ , which reproduces according to $(\xi_{\varnothing}, W_{\varnothing})$ . Thereafter, at times corresponding to points in $\xi_{\varnothing}$ , descendants of $\varnothing$ are formed, which in turn produce offspring according to the same law. A crucial aspect of the study of CMJ processes is the characteristics $\phi_{x}$ associated to each element $x \in \mathcal{U}$ . For $x \in \mathcal{U}$ , let $\mathcal{U}_{x} \,:\!=\, \left\{xu\,:\, u \in \mathcal{U}\right\}$ . Then the processes $\phi_{x}$ are identically distributed, non-negative stochastic processes on the space $(\Omega, \Sigma, \mathbb{P})$ associated with individuals x, which may depend on $(\xi_{z}, W_{z})_{z \in \mathcal{U}_{x}}$ . Intuitively, these are processes that track ‘characteristics’ not only of the individual x, but also of its potential offspring $\{xy\,:\, y \in \mathcal{U}\}$ . We then define the general branching process counted with characteristic as

\begin{equation*}Z^{\phi}(t) \,:\!=\, \sum_{x \in \mathcal{U}\,:\, \sigma_x \leq t} \phi_x\big(t-\sigma_x\big);\end{equation*}

thus this function keeps a ‘score’ of characteristics of individuals in the family tree associated with the process up to time t.

Let $\nu$ be the intensity measure of $\xi$ , that is, $\nu(B) \,:\!=\, {{\mathbb{E}\!\left[{\xi(B)} \right]}}$ for measurable sets $B \subseteq \mathbb{R}_{+}$ . A crucial parameter in the study of CMJ processes is the Malthusian parameter $\alpha$ , which is defined as the solution (if it exists) of

\begin{equation*}{{\mathbb{E}\!\left[{\int_{0}^{\infty} e^{-\alpha u} \xi({\text{d}} u)} \right]}} = 1.\end{equation*}

Assume that $\nu$ is not supported on any lattice, i.e., for any $h > 0$ , ${{\text{Supp} \!\left( {\nu} \right)}} \subsetneq \{ 0, h, 2h, \ldots\}$ , and that the first moment of $e^{-\alpha u} \nu({\text{d}} u)$ is finite, i.e., $\int_{0}^{\infty} u e^{-\alpha u} \nu({\text{d}} u) < \infty$ . Nerman [Reference Nerman30] proved the following theorem.

Theorem 2.3. ([Reference Nerman30, Theorem 6.3]) Suppose that there exists $\lambda < \alpha$ satisfying

(16) \begin{equation} {{\mathbb{E}\!\left[{\int_{0}^{\infty} e^{-\lambda s} \xi({\text{d}} s)} \right]}} < \infty.\end{equation}

Then, for any two càdlàg characteristics $\phi^{(1)}, \phi^{(2)}$ such that ${{\mathbb{E}\!\left[{\sup_{t \geq 0} e^{-\lambda t} \phi^{(i)}(t)} \right]}} < \infty, \; i=1, 2,$ we have

\begin{equation*} \lim_{t \to \infty} \frac{Z^{\phi^{(1)}}(t)}{Z^{\phi^{(2)}}(t)} = \frac{\int_0^\infty \text e^{-\alpha s}{{\mathbb{E} \!\left[{\phi^{(1)}(s)} \right]}} {\text{d}} s}{\int_0^\infty \text e^{-\alpha s}{{\mathbb{E}\!\left[{\phi^{(2)}(s)} \right]}} {\text{d}} s}, \end{equation*}

almost surely on the event $\{\left|\mathbb{T}_{t}\right| \rightarrow \infty\}$ .

Recall the definition of $\rho$ as the point process associated with the jumps in the process Y defined in (9). Then the continuous-time model outlined in Section 2.1 is a CMJ process having $\rho$ as its associated random point process and weight W. In this case, the Malthusian parameter is given by $\alpha$ in (12), and moreover, Condition C1 implies that the first moment $\int_{0}^{\infty} t e^{-\alpha t} \hat{\rho}_{\mu}({\text{d}} t)$ is finite.

Theorem 2.1 is now an immediate application of Theorem 2.3.

Proof of Theorem 2.1. Consider the continuous-time branching process outlined in Section 2.1 and denote by $\sigma^{\prime}_1 < \sigma^{\prime}_2 \cdots$ the times of births of individuals in the process. Then ${\mathcal{T}}_n$ has the same distribution as the family tree $\mathbb{T}_{\sigma^{\prime}_n}$ . For any measurable set $B \subseteq \mathbb{R}$ , define the characteristics $\phi^{(1)}(t) = \mathbf{1}_{\left\{Y(t) = k\ell, W \in B\right\}}$ and $\phi^{(2)}(t) = \mathbf{1}_{\{t \geq 0\}}$ , where W denotes the weight of the process Y. Note that $Z^{\phi^{(1)}}(t)$ is the number of individuals with $k\ell$ offspring and weight belonging to B up to time t, while $Z^{\phi^{(2)}}(t) = |\mathbb{T}_{t}|$ . Thus,

\begin{equation*}\lim_{t \to \infty} \frac{Z^{\phi^{(1)}}(t)}{Z^{\phi^{(2)}}(t)} = \lim_{t \to \infty} \frac{N_{k}(t, B)}{\ell t}.\end{equation*}

Note that both $\phi^{(1)}(t)$ and $\phi^{(2)}(t)$ are càdlàg and bounded, and moreover, Condition C1 implies that (16) is satisfied. In addition, the assumption that $f(0,W) > 0$ almost surely implies that $\left|\mathbb{T}_{t}\right| \rightarrow \infty$ almost surely. Thus, by applying Theorem 2.3, we have

(17) \begin{align} \lim_{t \to \infty} \frac{Z^{\phi^{(1)}}(t)}{Z^{\phi^{(2)}}(t)} &= \alpha \int_0^\infty \text e^{-\alpha s}{{\mathbb{E} \!\left[{\mathbf{1}_{\left\{Y(s) = k\ell, W \in B\right\}}} \right]}} {\text{d}} s \nonumber\\&= {{\mathbb{E}\!\left[{{{\mathbb{E}_{W} \!\left[{ \left(e^{-\alpha \tau_{k}} - e^{-\alpha\tau_{k+1}}\right)} \right]}}\mathbf{1}_{B}(W)} \right]}},\end{align}

where the last equality follows from Fubini’s theorem and we recall that $\tau_{k}$ is the time of the kth event in the process $Y_{W}(t)$ . Now, since, when $W = w$ , $\tau_k$ is distributed as a sum of independent exponentially distributed random variables with rates $f(0,w), f(1,w) \ldots$ , we have

(18) \begin{equation} {{\mathbb{E}\!\left[{{{\mathbb{E}_{W} \!\left[{e^{-\alpha \tau_{k}}} \right]}}\mathbf{1}_{B}(W)} \right]}} = {{\mathbb{E}\!\left[{\left(\prod_{i=0}^{k-1} \frac{f(i,W)}{f(i,W) + \alpha} \right)\mathbf{1}_{B}(W)} \right]}}.\end{equation}

The result follows from combining (17) and (18).

2.3. A strong law for the partition function

We can also apply Theorem 2.3 to show that the Malthusian parameter $\alpha$ emerges as the almost sure limit of the partition function, under certain conditions on the fitness function f.

Theorem 2.4. Let $(\mathcal{T}_{t})_{t \geq 0}$ be a ${{\left({\mu, f, \ell}\right)}}$ -RIF tree satisfying C1 with Malthusian parameter $\alpha$ . Moreover, assume that there exists a constant $C < \alpha$ and a non-negative function $\varphi$ with ${{\mathbb{E} \!\left[{\varphi(W)} \right]}} < \infty$ such that, for all $k \in \mathbb{N}_{0}$ , $f(k,W) \leq C k + \varphi(W)$ almost surely. Then, almost surely,

\begin{equation*}\frac{{\mathcal{Z}}_{t}}{t} \xrightarrow{t \to \infty} \alpha.\end{equation*}

To apply Theorem 2.3, we need to bound ${{\mathbb{E}\!\left[{\sup_{t \geq 0} e^{-\lambda t} \phi^{(1)}(t)} \right]}}$ for an appropriate choice of $\lambda < \alpha$ and characteristic $\phi^{(1)}$ that tracks the evolution of the partition function associated with the process. In order to do so, using the assumptions on f(i, W), we will couple the process Y defined in (9) with an appropriate pure birth process $(\mathcal{Y}(t))_{t\geq 0}$ (Lemma 2.3) and apply Doob’s maximal inequality to a martingale associated with $(\mathcal{Y}(t))_{t \geq 0}$ (Lemma 2.2). As we will see, our choice of $\lambda$ will be given by C, and this is the reason for the assumption that $C < \alpha$ in Theorem 2.4.

In order to define $\mathcal{Y}(t)$ , first sample a weight W and set $\mathcal{Y}(0) = 0$ . Then, if $\mathbb{P}_{w}$ denotes the probability measure associated with the process when the weight is w, define the rates so that

\begin{equation*} {{\mathbb{P}_{w} \!\left({\mathcal{Y}(t + h) = k + 1 \; | \; \mathcal{Y}(t) = k} \right)}} = (C k + \varphi(w)) h + o(h). \end{equation*}

We also let $\mathcal{Y}_{w}$ denote the process with the same transition rates, but deterministic weight w.

It will be beneficial to state a more general result, about pure birth processes $\left(\mathcal{X}(t)\right)_{t\geq 0}$ with linear rates, from the paper by Holmgren and Janson [Reference Holmgren and Janson20]. For brevity, we adapt the notation and only include some specific statements from both theorems.

Lemma 2.1. ([Reference Holmgren and Janson20, Theorem A.6 and Theorem A.7]) Let $\left(\mathcal{X}(t)\right)_{t\geq 0}$ be a pure birth process with $\mathcal{X}(0) = x_0$ and rates such that

\begin{equation*}{{\mathbb{P}\!\left({\mathcal{X}(t + h) = k + 1 \; | \; \mathcal{X}(t) = k}\right)}} = (c_1 k + c_2) h + o(h),\end{equation*}

for some constants $c_1, c_2 > 0$ . Then, for each $t \geq 0$ ,

(19) \begin{equation} {{\mathbb{E}\!\left[{\mathcal{X}(t)} \right]}} = \left(x_0 + \frac{c_2}{c_1}\right)e^{c_1 t} - \frac{c_2}{c_1}.\end{equation}

Moreover, if $x_0 = 0$ , the probability generating function is given by

(20) \begin{equation} {{\mathbb{E}\!\left[{z^{\mathcal{X}(t)}} \right]}} = \left(\frac{e^{-c_1t}}{1-z\left(1 - e^{-c_1t}\right)}\right)^{c_2/c_1}.\end{equation}

Finally, we will require Lemma 2.2 and Lemma 2.3.

Lemma 2.2. For any $w > 0$ , the process $(e^{-C t}\left(\mathcal{Y}_{w}(t) + \varphi(w)/C\right))_{t \geq 0}$ is a martingale with respect to its natural filtration $(\mathcal{F}_t)_{t\geq 0}$ . Moreover,

\begin{equation*}{{\mathbb{E}\!\left[{\sup_{t \geq 0} \left(e^{-C t}\mathcal{Y}(t) \right)} \right]}} < \infty.\end{equation*}

Proof. The process $(\mathcal{Y}_{w}(t))_{t \geq 0}$ is a pure birth process satisfying the assumptions of Lemma 2.1, with $c_1 = C$ and $c_2 = \varphi(w)$ . Therefore, by (19) and the Markov property, for any $t > s >0$ we have

\begin{equation*}{{\mathbb{E}\!\left[{\mathcal{Y}_{w}(t) \; | \; \mathcal{F}_{s}} \right]}} = {{\mathbb{E}\!\left[{\mathcal{Y}_{w}(t) \; | \; \mathcal{Y}_{w}(s)} \right]}} =\left(\mathcal{Y}_{w}(s) + \frac{\varphi(w)}{C} \right)e^{C(t-s)} - \frac{\varphi(w)}{C},\end{equation*}

which implies the martingale statement.

Moreover, applying (20) for the probability generating function, differentiating twice and evaluating at $z = 1$ , we obtain

\begin{equation*}{{\mathbb{E}\!\left[{\mathcal{Y}_{w}(t)\left(\mathcal{Y}_{w}(t) -1 \right)} \right]}}= \frac{\varphi(w)\left(C + \varphi(w)\right)}{C^2} \big(e^{C t} -1\big)^2,\end{equation*}

and thus after some manipulations, we find that for all $t \geq 0$

\begin{equation*}{{\mathbb{E}\!\left[{e^{-2Ct}\left(\mathcal{Y}_{w}(t) + \varphi(w)/C\right)^2} \right]}} \leq {\frac{\varphi(w)^2}{C^2} + \frac{\varphi(w)}{C}\big(1 - e^{-Ct}\big)}.\end{equation*}

Combining this $L^2$ quadratic bound with Doob’s maximal inequality, we have

\begin{align*}{{\mathbb{E}\!\left[{\sup_{t \geq 0} \left(e^{-C t}\mathcal{Y}_w(t) \right)} \right]}} & \leq{{\mathbb{E}\!\left[{\sup_{t \geq 0} \left(e^{-C t}\left(\mathcal{Y}_w(t) + \varphi(w)/C\right)\right)} \right]}}\\ & \leq A + B\varphi(w),\end{align*}

for constants A, B depending only on C. Thus,

\begin{equation*}{{\mathbb{E}\!\left[{\sup_{t \geq 0} \left(e^{-C t}\mathcal{Y}(t) \right)} \right]}} = {{\mathbb{E}\!\left[{\sup_{t \geq 0} \left(e^{-C t}\mathcal{Y}_{W}(t) \right)} \right]}} \leq A + B{{\mathbb{E}\!\left[{\varphi(W)} \right]}} < \infty.\end{equation*}

Lemma 2.3. Recall the definition of Y in (9) and assume that there exists a constant $C < \alpha$ and a non-negative function $\varphi$ with ${{\mathbb{E}\!\left[{\varphi(W)} \right]}} < \infty$ such that, for all $k \in \mathbb{N}_{0}$ , $f(k,W) \leq C k + \varphi(W)$ almost surely. Then there exists a coupling $(\hat{Y}(t),\hat{\mathcal{Y}}(t))_{t \geq 0}$ of $(Y(t))_{t \geq 0}$ and $(\mathcal{Y}(t))_{t \geq 0}$ such that, for all $t \geq 0$ ,

\begin{equation*}\hat{Y}(t) \leq \ell \cdot \hat{\mathcal{Y}}(t). \end{equation*}

In the following proof, we denote by ${\text{Exp}\!\left( {r} \right)}$ the exponential distribution with parameter r.

Proof. First, we sample $\hat{W}$ from $\mu$ and use this as a common weight for $\hat{Y}$ and $\hat{\mathcal{Y}}$ . Now, let $\left(\varsigma_{i}\right)_{i \geq 0}$ be independent ${\text{Exp}\!\left( {f(i, \hat{W})} \right)}$ -distributed random variables. Then, for all $k > 0$ , set $\hat{\tau}_{k} = \sum_{i=0}^{k-1} \varsigma_{i}$ and

\begin{equation*}\hat{Y}(t) = \sum_{k=1}^{\infty} k \ell \mathbf{1}_{\left\{\hat{\tau}_{k} \leq t < \hat{\tau}_{k+1}\right\}}.\end{equation*}

The $\varsigma_{i}$ can be interpreted as the intermittent time between jumps from state i to $i+\ell$ . For all $t> 0$ construct the jump times of $(\hat{\mathcal{Y}}(t))_{t \geq 0}$ iteratively as follows:

• Note that by assumption $f(0, \hat{W}) \leq \varphi(\hat{W})$ . Let $e_{0} \sim {\text{Exp}\!\left( {\varphi(\hat{W}) - f(0,\hat{W})} \right)}$ and set $\varsigma^{\prime}_{0} = \min\!\left\{e_{0}, \varsigma_{0}\right\}$ . We may interpret $\varsigma^{\prime}_{0}$ as the time for $\hat{\mathcal{Y}}$ to jump from 0 to 1.

• Given $\varsigma^{\prime}_{0}, \ldots, \varsigma^{\prime}_{j}$ , let $q_{j} \,:\!=\, \sum_{i=0}^{j}\varsigma^{\prime}_{i}$ and define $m_{j} \,:\!=\, \hat{Y}(q_{j})/\ell$ , i.e., the value of $\hat{Y}/\ell$ once $\hat{\mathcal{Y}}$ has reached $j+1$ . Assume inductively that $m_{j} \leq j+1$ and set

  \begin{equation*}e_{j + 1} \sim {\text{Exp}\!\left( {C(j + 1) +\varphi(\hat{W}) - f\big(m_{j}, \hat{W}\big)} \right)} \quad \text{ and } \quad \varsigma^{\prime}_{j+1} = \min\!\left\{e_{j}, \varsigma_{m_j} \right\}.\end{equation*}

Observe that, since $\varsigma^{\prime}_{j+1} \leq \varsigma_{m_j + 1}$ , we have $m_{j+1} \leq j+2$ , so we may iterate this procedure.

It is clear that $(\hat{Y}(t))_{t \geq 0}$ is distributed like $(Y(t))_{t \geq 0}$ , and using the properties of the exponential distribution one readily confirms that $(\hat{\mathcal{Y}}(t))_{t \geq 0}$ is distributed like $(\mathcal{Y}(t))_{t\geq 0}$ . Finally, the desired inequality follows from the fact that $\hat{\mathcal{Y}}(t)$ always jumps before or at the same time as $\hat{Y}(t)$ .

Proof of Theorem 2.4. Consider the continuous-time embedding of the ${{\left({\mu, f, \ell}\right)}}$ -RIF tree and define the characteristics $\phi^{(1)}(t) \,:\!=\, \sum_{k=0}^{\infty} f(k,W) \mathbf{1}_{\{Y(t) = k\ell\}}$ and $\phi^{(2)}(t) \,:\!=\, \mathbf{1}_{\{t \geq 0\}}$ . Recall that we denote by $(\tau_i)_{i \geq 1}$ the times of the jumps in Y and that, for all $k \geq 0$ , $f(k,W) \leq C k + \varphi(W)$ . Then, by Lemma 2.3, Lemma 2.2, and the assumptions of the theorem,

\begin{equation*}{{\mathbb{E}\!\left[{\sup_{t \geq 0} \left(e^{-C t} \phi^{(1)}(t)\right)} \right]}} \stackrel{\text{Lem.}\ 2.3}{\leq} {{\mathbb{E}\!\left[{\sup_{t \geq 0} \left(e^{-C t} \left(C \mathcal{Y}_{W}(t) + \varphi(W)\right) \right)} \right]}} \stackrel{\text{Lem.}\ 2.2}{<} \infty.\end{equation*}

Now, in this case $Z^{\phi^{(1)}}(t)$ is the total sum of fitnesses of individuals born up to time t, while $Z^{\phi^{(2)}}(t) = |\mathcal{T}_{t}|$ . Thus, by Theorem 2.3 and Fubini’s theorem in the second equality, almost surely we have

(21) \begin{align} \lim_{n \to \infty} \frac{\mathcal{Z}_{n}}{\ell n} & = \alpha \int_{0}^{\infty} e^{- \alpha s} {{\mathbb{E}\!\left[{\sum_{k=0}^{\infty} f(k,W) \mathbf{1}_{\{Y(s) = k\ell\}}}\right]}} {\text{d}} s ={{\mathbb{E}\!\left[{\sum_{k=0}^{\infty} f(k,W) \left(e^{-\alpha \tau_{k}} - e^{-\alpha \tau_{k+1}}\right)} \right]}} \end{align}

\begin{align} \nonumber & = {{\mathbb{E}\!\left[{\sum_{k=1}^{\infty} \frac{\alpha f(k,W)}{f(k,W) + \alpha}\prod_{i=0}^{k-1} \frac{f(i,W)}{f(i,W) + \alpha}} \right]}}.\end{align}

Now, recall that by (12) we have

\begin{equation*}{{\mathbb{E}\!\left[{\sum_{k=1}^{\infty} \frac{f(k,W)}{f(k,W) + \alpha}\prod_{i=0}^{k-1} \frac{f(i,W)}{f(i,W) + \alpha}} \right]}} = \frac{1}{\ell},\end{equation*}

and combining this with (21) proves the result.

2.4. Examples of applications of Theorem 2.1

2.4.1. Weighted Cayley trees

Consider the model where $f(k,W) = 0$ for $k \geq 1$ and $f(0,W) = g(W)$ . Thus, at each step, a vertex with degree 0 is chosen and produces $\ell$ children, and thus this model produces an $(\ell + 1)$ -Cayley tree, i.e., a tree in which each node that is not a leaf has degree $\ell + 1$ . Without loss of generality, by considering the pushforward of $\mu$ under g if necessary, we may assume that $g(W) = W$ . In this case, $\hat{\rho}_{\mu}(\lambda) = \ell \cdot {{\mathbb{E}\!\left[{\frac{W}{W + \lambda}} \right]}}$ and thus C1 is satisfied as long as $\ell \geq 2$ . Thus, $p^{\alpha}_{k}(B) = 0$ for all $k \geq 2$ , and

\begin{equation*}p_0(B) = {{\mathbb{E}\!\left[{\frac{\alpha}{W + \alpha}\mathbf{1}_{B}(W)} \right]}}, \quad p_1(B) = {{\mathbb{E} \!\left[{\frac{W}{W + \alpha} \mathbf{1}_{B}(W)} \right]}}.\end{equation*}

This rigorously confirms a result of Bianconi [Reference Bianconi4]. Note, however, that in [Reference Bianconi4] $\alpha$ is described as the almost sure limit of the partition function, and we may only apply Theorem 2.4 under the assumption that ${{\mathbb{E} \!\left[W \right]}}< \infty$ .

In the notation of [Reference Bianconi4], the weights W are called ‘energies’, using the symbol $\epsilon$ , the function $g(\epsilon) \,:\!=\, e^{\beta \epsilon}$ , where $\beta > 0$ is a parameter of the model, and $\alpha \,:\!=\, e^{\beta \mu_{F}}$ is described as the limit of the partition function. Thus, the proportion of vertices with out-degree 0 with ‘energy’ belonging to some measurable set B is

\begin{equation*}{{\mathbb{E}\!\left[{\frac{1}{e^{\beta (\epsilon - \mu_{F})} + 1}\mathbf{1}_{B}(W)} \right]}}, \end{equation*}

which is known as a Fermi–Dirac distribution in physics.

2.4.2. Weighted random recursive trees

In the case that $f(k,W) = W$ , we obtain a model of weighted random recursive trees with independent weights and C1 is satisfied with $\alpha = {{\mathbb{E}\!\left[{W} \right]}}$ provided ${{\mathbb{E}\!\left[{W} \right]}} < \infty$ . Theorem 2.1 then implies that

\begin{equation*}\frac{N_{k}(t, B)}{\ell t} \xrightarrow{t \to \infty} {{\mathbb{E}\!\left[{\frac{\ell {{\mathbb{E}\!\left[{W} \right]}} W^k}{\left(W + \ell{{\mathbb{E}\!\left[{W} \right]}}\right)^{k+1}} \mathbf{1}_{B}(W)} \right]}},\end{equation*}

almost surely. This was observed in the case $\ell = 1$ by the authors of [Reference Fountoulakis, Iyer, Mailler and Sulzbach17, Proposition 3]. Note also that in this case Theorem 2.4 coincides with the usual strong law of large numbers.

The weighted random recursive tree has a natural generalisation to affine fitness functions. This is the topic of the next section.

3.1. When the GPAF-tree satisfies Condition C1

In the context of the GPAF-tree, Condition C1 states that there exists $\lambda > 0$ such that

\begin{equation*}m(\lambda, \mathbb{R}_{+}) = \ell \cdot {{\mathbb{E}\!\left[{\sum_{n=1}^{\infty} \prod_{i=0}^{n-1} \frac{g(W)i +h(W)}{g(W)i + h(W) + \lambda}} \right]}} > 1.\end{equation*}

First recall the definition of the birth process Y from (9) in Section 2, with $f(k,{W}) = g(W)k + h(W)$ . By applying (19) from Lemma 2.1 and the initial condition $Y(0) = 0$ , for any $w \in \mathbb{R}_{+}$ we have

\begin{equation*}{{\mathbb{E}_{w} \left[{Y(t)} \right]}} = \left(\frac{h(w)}{g(w)}\right) e^{\ell g(w)t} - \frac{h(w)}{g(w)}.\end{equation*}

Now, (10) and (11) in Section 2 showed that

(22) \begin{equation} \ell \cdot \sum_{n=1}^{\infty} \prod_{i=0}^{n-1} \frac{g(W)i +h(W)}{g(W)i + h(W) + \lambda} = \int_{0}^{\infty} \lambda e^{-\lambda s} {{\mathbb{E}_{w} \!\left[{Y(s)} \right]}} {\text{d}} s =\begin{cases}\frac{h(w)}{\lambda/\ell - g(w)} & \text{if } \lambda/ \ell > g(w); \\[5pt]\infty & \text{otherwise}.\end{cases}\end{equation}

For a measurable function $g\,:\,\mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ we define $\text{ess sup}{(g)}$ by

\begin{equation*} \text{ess sup}{(g)} \,:\!=\, \inf\left\{a \in \mathbb{R}_{+}\,:\, \mu\!\left(\left\{x \,:\,g(x) > a\right\}\right) = 0\right\}.\end{equation*}

Therefore by (22), for $\lambda \geq \ell \cdot \text{ess sup}{(g)}$ we have

\begin{equation*}m(\lambda, \mathbb{R}_{+}) = {{\mathbb{E}\!\left[{\frac{h(W)}{\lambda/\ell - g(W)}} \right]}},\end{equation*}

while if $\lambda < \ell \cdot \text{ess sup}{(g)}$ we have $m(\lambda, \mathbb{R}_{+}) = \infty$ . Thus, Condition C1 is satisfied if $\text{ess sup}{(g)} < \infty$ , ${{\mathbb{E} \!\left[{h(W)} \right]}} < \infty$ , and for some $\lambda \geq \ell \cdot \text{ess sup}{(g)}$

\begin{equation*} 1 < {{\mathbb{E}\!\left[{\frac{h(W)}{\lambda/\ell - g(W)}} \right]}} < \infty.\end{equation*}

As a result, the Malthusian parameter $\alpha$ appearing in Condition C1 is given by the unique $\alpha > 0$ such that

(23) \begin{equation} {{\mathbb{E}\!\left[{\frac{h(W)}{\alpha/\ell - g(W)}} \right]}} = 1.\end{equation}

Note that the parameter $\ell$ in the model has the effect of re-scaling the Malthusian parameter $\alpha$ . Also, since $\alpha \geq \ell \cdot \text{ess sup}(g)$ , if ${{\mathbb{E}\!\left[{h(W)} \right]}} < \infty$ , Theorem 2.4 applies and $\alpha$ may also be interpreted as the almost sure limit of the partition function associated with the process. Now, in the context of this model, the limiting value $p^{\alpha}_{k}({\cdot})$ from Theorem 2.1 is such that

(24) \begin{equation} p^{\alpha}_{k}(B) = {{\mathbb{E}\!\left[{\frac{\alpha}{g(W)k + h(W) + \alpha} \prod_{i=0}^{k-1} \frac{g(W)i + h(W)}{g(W)i + h(W) + \alpha} \mathbf{1}_{B}(W)} \right]}}.\end{equation}

Now, recall Stirling’s approximation, which states that

(25) \begin{equation} \Gamma(z) = \left(1 + O(1/z)\right)z^{z - \frac{1}{2}} e^{-z}\end{equation}

as $z \to \infty$ . If $g(W) > 0$ on B, by dividing the numerator and denominator of terms inside the product in (24), we obtain a ratio of gamma functions. Thus, by applying Stirling’s approximation, on any measurable set B on which g, h are bounded, we have

\begin{equation*}p^{\alpha}_{k}(B) = \left(1 + O(1/k)\right){{\mathbb{E}\!\left[{c_{B}k^{-\left(1 + \frac{\alpha}{g(W)}\right)} \mathbf{1}_{B}(W)} \right]}},\end{equation*}

where $c_{B}$ , which comes from the term outside the product in (24), depends on g and h but not k. Thus, the distribution of $(p^{\alpha}_{k}(B))_{k \in \mathbb{N}_{0}}$ follows what one might describe as an ‘averaged’ power law. Moreover, in the case that $\ell = 1$ , we have $\alpha \geq \text{ess sup}(g)$ ; thus,

\begin{equation*}{{\mathbb{E}\!\left[{c_{B}k^{-\left(1 + \frac{\alpha}{g(W)}\right)} \mathbf{1}_{B}(W)} \right]}} \geq c^{\prime} k^{-2}\end{equation*}

for some $c^{\prime} > 0$ . It has been observed that real-world complex networks have power law degree distributions where the observed power law exponent lies between 2 and 3 (see, for example, [Reference Van der Hofstad37]). Note that by (23), $\alpha$ depends on both h and g, so that keeping g fixed and making h smaller has the effect of reducing the exponent of the power law.

In the remainder of this section we set $\ell =1$ , for brevity. The arguments may be adapted in a similar manner to the case $\ell >1$ .

3.2. A condensation phenomenon in the GPAF-tree when Condition C1 fails Condition C1 fails

Recall that, in the GPAF-tree, if $\lambda \geq \text{ess sup}{(g)}$ we have

\begin{equation*} m(\lambda, \mathbb{R}_{+}) = {{\mathbb{E}\!\left[{\frac{h(W)}{\lambda - g(W)}} \right]}},\end{equation*}

and if $\lambda < \text{ess sup}{(g)}$ we have $m(\lambda, \mathbb{R}_{+}) = \infty$ . If we define

\begin{equation*} \Lambda \,:\!=\, \left\{\lambda > 0\,:\, m(\lambda, \mathbb{R}_{+}) < \infty \right\},\end{equation*}

in this subsection, we consider the case that the GPAF-tree fails to satisfy Condition C1 by having $m(\lambda, \mathbb{R}_{+}) \leq 1$ for all $\lambda \in \Lambda$ . We show that in this case the GPAF-tree satisfies a formula for the degree distribution of the same form as (3). Moreover, if $\lambda^*\,:\!=\, \inf{\left(\Lambda\right)}$ and $m(\lambda^*, \mathbb{R}_{+}) < 1$ , this model exhibits a condensation phenomenon, as described in Theorem 3.1. We remark that such results have been proved for the case of the preferential attachment tree with multiplicative fitness, i.e., the case $h \equiv g$ , in [Reference Dereich and Ortgiese14], in a more general framework; that is to say, encompassing other models apart from a tree.

In Section 3.2.1 we state our main result, Theorem 3.1; we discuss interesting implications in Section 3.2.2. In Section 3.2.3 we state and prove Lemma 3.1, which is the crucial tool used in proofs of the theorem. The proof of Theorem 3.1 is deferred to Section 3.2.4.

3.2.1. Theorem 3.1: condensation in the GPAF-tree

Our main result in this subsection is the following theorem, which demonstrates the possibility of condensation in this model. Recall that in this section, we have $\lambda^* = \text{ess sup}{(g)}$ . We then define the following family of sets of positive $\mu$ -measure, depending on a parameter $\varepsilon > 0$ :

(26) \begin{equation} \mathcal{M}_{\varepsilon} \,:\!=\, \left\{x\,:\, g(x) \geq \lambda^{*} - \varepsilon\right\}.\end{equation}

Theorem 3.1. Suppose ${\mathcal{T}} = \left(\mathcal{T}_{t}\right)_{t \geq 0}$ is a GPAF-tree, with associated functions g, where g is bounded, ${{\mathbb{E}\!\left[{h(W)} \right]}} < \infty$ , and Condition C1 fails. Then we have the following assertions:

1. 1. For any measurable set B such that for some $\varepsilon > 0$ we have $B \subseteq \mathcal{M}^{c}_{\varepsilon}$ ,

   \begin{equation*} \frac{\Xi(t,B)}{\ell t} \xrightarrow{t \to \infty} {{\mathbb{E}\!\left[{\frac{h(W)}{\lambda^{*} - g(W)} \mathbf{1}_{B}(W)} \right]}}, \quad {almost\ surely.}\end{equation*}
   In particular, if
   \begin{equation*}{{\mathbb{E}\!\left[{\frac{h(W)}{g({w^{*}}) - g(W)}} \right]}} < 1,\end{equation*}
   for $\varepsilon > 0$ sufficiently small we have
   \begin{equation*}\frac{\Xi(t,\mathcal{M}_{\varepsilon})}{\ell t} \xrightarrow{t \to \infty} 1 - {{\mathbb{E}\!\left[{\frac{h(W)}{\lambda^{*} - g(W)} \mathbf{1}_{\mathcal{M}^{c}_{\varepsilon}}(W)} \right]}} > {{\mathbb{E}\!\left[{\frac{h(W)}{\lambda^{*} - g(W)} \mathbf{1}_{\mathcal{M}_{\varepsilon}}(W)} \right]}} = m(\lambda^*, \mathcal{M}_{\varepsilon}),\end{equation*}
   so that this model exhibits a condensation phenomenon, as described before Conjecture 1.1 in Section 1.3.

2. 2. For any measurable set $B \subseteq \mathbb{R}_{+}$ , almost surely we have

   \begin{equation*} \frac{N_{k}(t,B)}{t} \xrightarrow{t \to \infty} {{\mathbb{E}\!\left[{\frac{\lambda^{*}}{g(W)k + h(W) + \lambda^{*}} \prod_{i=0}^{k-1} \frac{g(W)i + h(W)}{g(W)i + h(W) + \lambda^{*}} \mathbf{1}_{B}(W)} \right]}} = p^{\lambda^{*}}_{k}(B).\end{equation*}

3. 3. The partition function satisfies

   \begin{equation*}\frac{{\mathcal{Z}}_{t}}{t} \xrightarrow{t \to \infty} \lambda^{*}, \quad \text{almost surely.} \end{equation*}

Suppose that ${w^{*}} \,:\!=\, \sup{\left({{\text{Supp} \!\left( {\mu} \right)}}\right)} < \infty$ , ${{\text{Supp} \!\left( {\mu} \right)}} \subseteq [0,{w^{*}}]$ , and g is increasing. Define the measure $\pi({\cdot})$ so that, for any measurable set B,

\begin{equation*}\pi(B) = {{\mathbb{E}\!\left[{\frac{h(W)}{g({w^{*}}) - g(W)} \mathbf{1}_{B}(W)} \right]}} + \left(1 - {{\mathbb{E}\!\left[{\frac{h(W)}{g({w^{*}}) - g(W)}} \right]}}\right)\delta_{{w^{*}}}(B).\end{equation*}

Then Assertion 1 of Theorem 3.1 leads to the following result.

Corollary 3.1. Under the above assumptions, with respect to the weak topology,

\begin{equation*}\frac{\Xi(t,\cdot)}{\ell t} \xrightarrow{t \to \infty} \pi({\cdot}), \quad {almost\ surely.}\end{equation*}

Remark 3.2. Corollary 3.1 is the form in which condensation results usually appear in the literature, showing that the limit of the sequence $\frac{\Xi(t,\cdot)}{\ell t}$ is no longer absolutely continuous with respect to $\mu$ . In this regard, Corollary 3.1 generalises the case $f(i,W) = (i+1)W$ which has already been proved in [Reference Borgs, Chayes, Daskalakis and Roch8].

Proof of Corollary 3.1. In view of the portmanteau theorem, it suffices to prove that, almost surely, for any open set O of $[0, {w^{*}}]$ we have

\begin{equation*}\liminf_{t \to \infty} \frac{\Xi(t, O)}{\ell t} \geq \pi(O).\end{equation*}

Now, it is well known that there exists a countable family of measurable sets $D_{1}, D_2, \ldots$ such that any open subset of $[0, {w^{*}}]$ may be expressed as a countable disjoint union of elements of this family. For example, one may take the set of all dyadic intervals, with endpoints of the form $j \cdot 2^{-n} {w^{*}}$ , $(j+1) \cdot 2^{-n} {w^{*}}$ , where $j, n \in \mathbb{N}_{0}$ . Fix such a family. Now, by Assertion 1 of Theorem 3.1, it is the case that, almost surely,

\begin{equation*} \lim_{t \to \infty} \frac{\Xi(t, S)}{t} = \pi(S) \quad \forall S \in \mathcal{C},\end{equation*}

where $\mathcal{C}$ is the countable collection of sets

\begin{equation*}\mathcal{C} \,:\!=\, \left\{D_{i} \cap \mathcal{M}^{c}_{1/j}, \, \mathcal{M}_{1/j}\,:\,i, j \in \mathbb{N}\right\}.\end{equation*}

Now, let O be an arbitrary open set. First, suppose that ${w^{*}} \notin O$ . Then, for a pairwise disjoint collection $D_{i_1}, D_{i_2}, \ldots$ such that $O = \bigcup_{\ell \in \mathbb{N}} D_{i_\ell}$ , for each $j, k \in \mathbb{N}$ we have

\begin{equation*}\liminf_{t \to \infty} \frac{\Xi(t,O)}{t} \geq \sum_{\ell = 1}^{k} \liminf_{t \to \infty} \frac{\Xi\big(t, D_{i_\ell} \cap \mathcal{M}^{c}_{1/j}\big)}{t} \geq \sum_{\ell = 1}^{k} \pi\big(D_{i_\ell} \cap \mathcal{M}^{c}_{1/j}\big).\end{equation*}

Taking limits in j and k, the right-hand side converges to $\pi(O)$ , as required. On the other hand, if ${w^{*}} \in O$ , since g is increasing, for each $\varepsilon > 0$ there exists $\delta = \delta(\varepsilon) > 0$ such that $\mathcal{M}_{\varepsilon} \subseteq [{w^{*}} - \delta, {w^{*}}]$ . Therefore, because O is open, for all j sufficiently large, we have $\mathcal{M}_{1/j} \subseteq O$ . But then, for a pairwise disjoint collection $D_{i_1}, D_{i_2}, \ldots$ such that $O = \bigcup_{\ell \in \mathbb{N}} D_{i_\ell}$ , we have

\begin{equation*}O = \mathcal{M}_{1/j} \cup \left(\bigcup_{\ell \in \mathbb{N}} D_{i_\ell} \cap \mathcal{M}^{c}_{1/j}\right).\end{equation*}

Therefore,

\begin{align*}\liminf_{t \to \infty} \frac{\Xi(t,O)}{t} &\geq \liminf_{t \to \infty} \frac{\Xi(t, \mathcal{M}_{1/j})}{t} + \sum_{\ell = 1}^{k} \liminf_{t \to \infty} \frac{\Xi\big(t, D_{i_\ell} \cap \mathcal{M}^{c}_{1/j}\big)}{t}\\ & \geq \pi(\mathcal{M}_{1/j}) + \sum_{\ell = 1}^{k} \pi\big(D_{i_\ell} \cap \mathcal{M}^{c}_{1/j}\big),\end{align*}

so that, by again taking limits in j and k, the right-hand side converges to $\pi(O)$ . The result follows.

3.2.3. A coupling lemma

In order to prove Theorem 3.1, we first prove an additional lemma. For each $\varepsilon > 0$ such that $\varepsilon < \lambda^*$ , let ${\mathcal{T}}^{+\varepsilon} = ({\mathcal{T}}^{+\varepsilon}_{t})_{t \geq 0}$ and ${\mathcal{T}}^{-\varepsilon} = ({\mathcal{T}}^{-\varepsilon}_{t})_{t \geq 0}$ denote GPAF-trees with the same set of weights, but with the function g modified to $g_{+\varepsilon}$ and $g_{-\varepsilon}$ respectively, on the set $\mathcal{M}_{\varepsilon}$ from (26), with

\begin{equation*}g_{+\varepsilon} \,:\!=\, g\mathbf{1}_{\mathcal{M}^{c}_{\varepsilon}} + \lambda^*\mathbf{1}_{\mathcal{M}_{\varepsilon}} \quad \text{and} \quad g_{-\varepsilon} \,:\!=\, g\mathbf{1}_{\mathcal{M}^{c}} + (\lambda^{*} - \varepsilon)\mathbf{1}_{\mathcal{M}_{\varepsilon}}.\end{equation*}

The motivation behind these choices of ${\mathcal{T}}^{+\varepsilon}$ and ${\mathcal{T}}^{-\varepsilon}$ is that, because ${\mathcal{T}}$ does not satisfy Condition C1, $g_{+\varepsilon}$ and $g_{-\varepsilon}$ attain their essential suprema on sets of positive measure. Thus, because ${{\mathbb{E}\!\left[{h(W)} \right]}} < \infty$ , by Remark 3.1 these auxiliary trees satisfy Condition C1, and we may apply the theorems from Section 2 with regard to these trees. Then, using the fact that these trees provide sufficiently good ‘approximations’ of the tree ${\mathcal{T}}$ , we may deduce our results by sending $\varepsilon$ to 0.

In this vein, let $N^{+\varepsilon}_{\geq k}(t,B)$ , $N_{\geq k}(t,B)$ , and $N^{-\varepsilon}_{\geq k}(t,B)$ denote the number of vertices with out-degree $\geq k$ and weight belonging to the set B in ${\mathcal{T}}^{+\varepsilon}_{t}$ , ${\mathcal{T}}_{t}$ , and ${\mathcal{T}}^{-\varepsilon}_t$ , respectively. In their respective trees, we also denote by ${\mathcal{Z}}^{+\varepsilon}_t$ , ${\mathcal{Z}}_t$ , and ${\mathcal{Z}}^{-\varepsilon}_t$ the partition functions at time t. Finally, for brevity, we write $f^{(+\varepsilon)}_{t}(v)$ , $f_{t}(v)$ , and $f^{(-\varepsilon)}_{t}(v)$ for the fitness of a vertex v at time t in each of these models. For example, $f_{t}(v) = g(W_v){\text{deg}^{+}({v, {\mathcal{T}}_{t}})} + h(W_v)$ .

Lemma 3.1. For every $\varepsilon > 0$ , there exists a coupling $(\hat{{\mathcal{T}}}^{+\varepsilon}, \hat{{\mathcal{T}}}, \hat{{\mathcal{T}}}^{-\varepsilon})$ of these processes such that, on the coupling, for all $t \in \mathbb{N}_0$ , the following hold:

1. 1. For all measurable sets $B \subseteq \mathcal{M}_{\varepsilon}^{c}$ we have $\Xi^{+\varepsilon}(t,B) \leq \Xi(t,B) \leq \Xi^{-\varepsilon}(t,B)$ .

2. 2. For all measurable sets $B \subseteq \mathcal{M}_{\varepsilon}^{c}$ and $k \in \mathbb{N}_0$ , we have

   \begin{equation*}N^{+\varepsilon}_{\geq k}(t,B) \leq N_{\geq k} (t,B) \leq N^{-\varepsilon}_{\geq k}(t,B).\end{equation*}

3. 3. We have the inequalities ${\mathcal{Z}}^{-\varepsilon}_{t} \leq {\mathcal{Z}}_t \leq {\mathcal{Z}}^{+\varepsilon}_t.$

Proof of Lemma 3.1. We construct the trees having the same sequence of weights $(W_{i})_{i \geq 0}$ , so that the dynamics of the models are only influenced by differences in the function g in the respective models. Thus, at time 0 each model consists of a single vertex labelled 0 with weight $W_0$ and having fitness given by $h(W_0)$ . Assume, that at the tth time-step,

\begin{equation*}(\hat{{\mathcal{T}}}^{+\varepsilon}_{n})_{0 \leq n \leq t} \sim ({\mathcal{T}}^{+\varepsilon}_{n})_{0 \leq n \leq t}, \quad (\hat{{\mathcal{T}}}_{n})_{0 \leq n \leq t} \sim ({\mathcal{T}}_{n})_{0 \leq n \leq t}, \quad \text{and} \quad (\hat{{\mathcal{T}}}^{-\varepsilon}_{n})_{0 \leq n \leq t} \sim ({\mathcal{T}}^{-\varepsilon}_{n})_{0 \leq n \leq t}.\end{equation*}

In addition, assume, by induction, that we have ${\mathcal{Z}}^{-\varepsilon}_{t} \leq {\mathcal{Z}}_t \leq {\mathcal{Z}}^{+\varepsilon}_t$ and for each vertex v with $W_{v} \in \mathcal{M}^{c}_{\varepsilon}$

(27) \begin{equation} {\text{deg}^{+}({v, \hat{{\mathcal{T}}}^{+\varepsilon}_t})} \leq {\text{deg}^{+}({v, \hat{{\mathcal{T}}}_{t}})} \leq {\text{deg}^{+}\big({v, \hat{{\mathcal{T}}}^{-\varepsilon}_t}\big)}.\end{equation}

Note that (27) and the fact that the trees have the same number of directed edges imply the first and the second assertions of the lemma up to time t. As a result, for each vertex v with $W_{v} \in \mathcal{M}^{c}_\varepsilon$ we have $f^{(+\varepsilon)}_t(v) \leq f_t(v) \leq f^{(-\varepsilon)}_{t}(v)$ . Now, for the $(t+1)$ th step, we do the following:

• Introduce a vertex $t+1$ with weight $W_{t+1}$ sampled independently from $\mu$ .

• Form $\hat{{\mathcal{T}}}^{-\varepsilon}_{t+1}$ by sampling the parent v of $t+1$ independently according to the law of $\mathcal{T}^{-\varepsilon}$ , i.e., with probability proportional to $f^{(-\varepsilon)}_t(v)$ . Then, in order to form $\hat{{\mathcal{T}}}_{t+1}$ , sample an independent uniformly distributed random variable $U_1$ on [0, 1].

• If

  \begin{equation*}U_1 \leq \frac{{\mathcal{Z}}^{-\varepsilon}_{t}f_{t}(v) }{{\mathcal{Z}}_t f^{(-\varepsilon)}_{t}(v)}\end{equation*}
  and $W_{v} \in \mathcal{M}_{\varepsilon}^{c}$ , select v as the parent of $t+1$ in $\hat{{\mathcal{T}}}_{t+1}$ as well.

• Otherwise, form $\hat{{\mathcal{T}}}_{t+1}$ by selecting the parent v ^′ of $t+1$ with probability proportional to $f_{t}(v^{\prime})$ out of all the vertices with weight $W_{v^{\prime}} \in \mathcal{M}_{\varepsilon}$ .

• Then form $\hat{{\mathcal{T}}}^{+\varepsilon}_{t+1}$ from $\hat{{\mathcal{T}}}_{t+1}$ in an identical manner to the way $\hat{{\mathcal{T}}}_{t+1}$ is formed from $\hat{{\mathcal{T}}}^{-\varepsilon}$ , with another, independent uniform random variable $U_2$ on [0, 1].

It is clear that $\hat{{\mathcal{T}}}^{-\varepsilon}_{t+1} \sim {\mathcal{T}}^{-\varepsilon}_{t+1}$ . On the other hand, in $\hat{{\mathcal{T}}}_{t+1}$ the probability of choosing a parent v of $t+1$ with weight $W_v \in \mathcal{M}_{\varepsilon}^{c}$ is

\begin{equation*}\frac{{\mathcal{Z}}^{-\varepsilon}_{t}f_{t}(v)}{{\mathcal{Z}}_t f^{(-\varepsilon)}_{t}(v)} \times \frac{f_t^{(-\varepsilon)}(v)}{{\mathcal{Z}}^{-\varepsilon}_t} = \frac{f_t(v)}{{\mathcal{Z}}_t},\end{equation*}

whilst the probability of choosing a parent v ^′ with weight $W_{v^{\prime}} \in \mathcal{M}_{\varepsilon}$ is

\begin{align*}& \frac{f_t(v^{\prime})}{\sum_{v \,:\,W_{v} \geq {w^{*}} - \varepsilon} f_{t}(v)} \left(\sum_{v\,:\, W_{v} < {w^{*}} - \varepsilon}\left(1- \frac{{\mathcal{Z}}^{-\varepsilon}_{t}f_{t}(v)}{{\mathcal{Z}}_t f^{(-\varepsilon)}_{t}(v)}\right) \frac{f^{(-\varepsilon)}_t(v)}{{\mathcal{Z}}^{-\varepsilon}_t}\right) \\ & \qquad + \frac{f_t(v^{\prime})}{\sum_{v\,:\,W_{v} \geq {w^{*}} - \varepsilon} f_{t}(v)} \left(\sum_{v\,:\,W_v \geq {w^{*}} - \varepsilon} \frac{f^{(-\varepsilon)}_t(v)}{{\mathcal{Z}}^{-\varepsilon}_t}\right)\\& = \frac{f_t(v^{\prime})}{\sum_{v \,:\,W_{v} \geq {w^{*}} - \varepsilon} f_{t}(v)}\left(\sum_{v} \frac{f^{(-\varepsilon)}_t(v)}{{\mathcal{Z}}^{-\varepsilon}_t}- \sum_{v\,:\, W_{v} < {w^{*}} - \varepsilon} \frac{f_{t}(v)}{{\mathcal{Z}}_t} \right)\\& = \frac{f_t(v^{\prime})}{\sum_{v \,:\,W_{v} \geq {w^{*}} - \varepsilon} f_{t}(v)} \left(1 - \frac{\sum_{v\,:\, W_{v} < {w^{*}} - \varepsilon}f_{t}(v)}{{\mathcal{Z}}_t}\right) = \frac{f_t(v^{\prime})}{{\mathcal{Z}}_t},\end{align*}

where we use the fact that $\sum_{v} f_{t}(v) = {\mathcal{Z}}_t$ . Thus, we have $\hat{{\mathcal{T}}}_{t+1} \sim {\mathcal{T}}_{t+1}$ . Moreover, either the same vertex is chosen as the parent of $t+1$ in both $\hat{{\mathcal{T}}}^{-\varepsilon}_{t+1}$ and $\hat{{\mathcal{T}}}_{t+1}$ , or a vertex of weight belonging to $\mathcal{M}_{\varepsilon}$ is chosen as the parent of $t+1$ in $\hat{\mathcal{T}}_{t+1}$ . This implies the left inequality in (27); in addition, when combined with the fact that $g_{-\varepsilon}(W_{t+1}) \leq g(W_{t+1})$ , it guarantees that ${\mathcal{Z}}^{-\varepsilon}_{t+1} \leq {\mathcal{Z}}_{t+1}$ . The proofs of the fact that $\hat{{\mathcal{T}}}^{+\varepsilon}_{t+1} \sim {\mathcal{T}}^{+\varepsilon}_{t+1}$ , the right inequality in (27), and the fact that ${\mathcal{Z}}_{t+1} \leq {\mathcal{Z}}^{+\varepsilon}_{t+1}$ are similar, so we may thus iterate the coupling.

3.2.4. Proof of Theorem 3.1

In order to prove Theorem 3.1, we use the auxiliary GPAF-trees ${\mathcal{T}}^{+\varepsilon}$ and ${\mathcal{T}}^{-\varepsilon}$ according to Lemma 3.1.

Proof of Theorem 3.1. For the first assertion, suppose that B is measurable, with $B \subseteq \mathcal{M}^{c}_{\varepsilon}$ . Then, if we define the corresponding quantities $\Xi^{+\varepsilon}(t,\cdot)$ , $\Xi^{-\varepsilon}(t,\cdot)$ associated with ${\mathcal{T}}^{+\varepsilon}$ and ${\mathcal{T}}^{-\varepsilon}$ , from the coupling in Lemma 3.1, we have

\begin{equation*}\frac{\Xi^{+\varepsilon}(t,B)}{t} \leq \frac{\Xi(t,B)}{t} \leq \frac{\Xi^{-\varepsilon}(t,B)}{t}.\end{equation*}

Recall that the auxiliary trees ${\mathcal{T}}^{+\varepsilon}$ and ${\mathcal{T}}^{-\varepsilon}$ have associated functions $g_{+\varepsilon}$ and $g_{-\varepsilon}$ which attain their maxima on a set of positive measure and thus satisfy Condition C1, with Malthusian parameters $\alpha^{(+\varepsilon)} > \lambda^{*}$ and $\alpha^{(-\varepsilon)} > \lambda^{*} - \varepsilon$ . Moreover, note that, by the definition of $g_{-\varepsilon}$ ,

\begin{align*}{{\mathbb{E}\!\left[{\frac{h(W)}{\lambda^{*} - g_{-\varepsilon}(W)}} \right]}} & \leq {{\mathbb{E}\!\left[{\frac{h(W)}{\lambda^* - g(W)}}\right]}} \leq 1,\end{align*}

so that, recalling (23), $\alpha^{(-\varepsilon)} \leq \lambda^*$ . As a result, $\lambda^* - \varepsilon < \alpha^{(-\varepsilon)} \leq \lambda^*$ , so $\lim_{\varepsilon \downarrow 0} \alpha^{(-\varepsilon)} = \lambda^*$ . Thus, by Lemma 3.1, Theorem 2.2, and dominated convergence, almost surely we have

\begin{equation*}\limsup_{t \to \infty} \frac{\Xi(t,B)}{t} \leq \lim_{\varepsilon \to 0} {{\mathbb{E}\!\left[{\frac{h(W)}{\alpha^{(-\varepsilon)} - g_{-\varepsilon}(W)} \mathbf{1}_{B}(W)} \right]}}= {{\mathbb{E}\!\left[{\frac{h(W)}{\lambda^{*} - g(W)} \mathbf{1}_{B}(W)} \right]}}.\end{equation*}

Now, we also have $\lim_{\varepsilon \to 0} \alpha^{(+\varepsilon)} = \lambda^{*}$ . Indeed, suppose for the sake of contradiction that $\lim_{\varepsilon \to 0} \alpha^{(+\varepsilon)} = \alpha^{\prime} > \lambda^{*}$ . Then, because ${{\mathbb{E}\!\left[{h(W)} \right]}} < \infty$ , by dominated convergence we have

\begin{equation*}1 = \lim_{\varepsilon \to 0} {{\mathbb{E}\!\left[{\frac{h(W)}{\alpha^{(+\varepsilon)} - g_{+\varepsilon}(W)}} \right]}} = {{\mathbb{E}\!\left[{\frac{h(W)}{\alpha^{\prime} - g(W)}} \right]}}.\end{equation*}

But then, (23) is satisfied for $\lambda$ such that $\lambda^{*} < \lambda < \alpha^{\prime}$ , contradicting the assumption that Condition C1 fails for ${\mathcal{T}}$ .

It follows that $\lim_{\varepsilon \to 0} \alpha^{(+\varepsilon)} = \lambda^{*}$ , and thus, by Lemma 3.1 and dominated convergence, almost surely we have

\begin{equation*}\limsup_{t \to \infty} \frac{\Xi(t,B)}{t} \leq \lim_{\varepsilon \to 0} {{\mathbb{E}\!\left[{\frac{h(W)}{\alpha^{(-\varepsilon)} - g_{-\varepsilon}(W)} \mathbf{1}_{B}(W)} \right]}}= {{\mathbb{E}\!\left[{\frac{h(W)}{\lambda^{*} - g(W)} \mathbf{1}_{B}(W)}\right]}}.\end{equation*}

The first assertion follows.

For the second assertion, given a measurable set B, for each $\varepsilon > 0$ , set $B^{\varepsilon}\,:\!=\, B \cap \mathcal{M}_{\varepsilon}$ . Then, by Lemma 3.1, almost surely we have

\begin{align*}\limsup_{t \to \infty} \frac{N_{\geq k} (t, B)}{t} & \leq \liminf_{\varepsilon \to 0} \left({{\mathbb{E}\!\left[{\prod_{i=0}^{k-1} \frac{g_{-\varepsilon}(W)i + h(W)}{g_{-\varepsilon}(W)i + h(W) + \alpha^{(-\varepsilon)}} \mathbf{1}_{B^{\varepsilon}}(W)} \right]}}+ \mu(\mathcal{M}_{\varepsilon})\right)\\ & = \liminf_{\varepsilon \to 0} {{\mathbb{E}\!\left[{\prod_{i=0}^{k-1} \frac{g(W)i + h(W)}{g(W)i + h(W) + \alpha^{(-\varepsilon)}} \mathbf{1}_{B^{\varepsilon}}(W)} \right]}}\\ & = {{\mathbb{E}\!\left[{\prod_{i=0}^{k-1} \frac{g(W)i + h(W)}{g(W)i + h(W) + \lambda^{*}}\mathbf{1}_{B}(W)} \right]}}.\end{align*}

Similarly, almost surely,

\begin{align*}\liminf_{t \to \infty} \frac{N_{\geq k} (t, B)}{t} & \geq \limsup_{\varepsilon \to 0} {{\mathbb{E}\!\left[{\prod_{i=0}^{k-1} \frac{g_{+\varepsilon}(W)i + h(W)}{g_{+\varepsilon}(W)i + h(W) + \alpha^{(+\varepsilon)}} \mathbf{1}_{B^{\varepsilon}}(W)}\right]}} \\ & = \limsup_{\varepsilon \to 0} \mathbb{E}\!\left[\prod_{i=0}^{k-1} \frac{g(W)i + h(W))}{g(W)i + h(W) + \alpha^{(+\varepsilon)}} \mathbf{1}_{B^{\varepsilon}}(W)\right] \\ & = \mathbb{E}\!\left[\prod_{i=0}^{k-1} \frac{g(W)i + h(W))}{g(W)i + h(W) + \lambda^{*}}\mathbf{1}_{B}(W)\right] .\end{align*}

Finally, for the last assertion, by Lemma 3.1, for each $t \in \mathbb{N}_0$ we have

\begin{equation*}\frac{{\mathcal{Z}}^{-\varepsilon}_t}{t} \leq \frac{{\mathcal{Z}}_t}{t} \leq \frac{{\mathcal{Z}}^{+\varepsilon}_t}{t}.\end{equation*}

Taking limits as t goes to infinity and applying Theorem 2.4, the result follows similarly to the previous assertions.

3.3. Degenerate degrees in the GPAF-tree when Condition C1 fails

In this subsection, we show that if the GPAF-tree fails to satisfy Condition ${\textbf{C1}}$ by having $m(\lambda, \mathbb{R}_{+}) = \infty$ for all $\lambda > 0$ , almost surely the proportion of vertices that are leaves tends to 1. Consequently, the limiting mass of edges ‘escapes to infinity’, as described in Theorem 3.2 below. Note that Condition C1 fails in this manner in the GPAF tree if $\text{ess sup}{(g)} = \infty$ or ${{\mathbb{E}\!\left[{h(W)} \right]}} = \infty$ . We remark that results similar to Theorem 3.2 have been shown in preferential attachment models with multiplicative fitness with $\mu$ having finite support [Reference Borgs, Chayes, Daskalakis and Roch8, Theorem 6] and preferential attachment models with additive fitness (the extreme disorder regime in [Reference Lodewijks and Ortgiese23, Theorem 2.6]). These cases correspond to $h(x) \equiv 0$ and $g(x) \equiv 1$ , respectively. In a similar vein to the start of Section 3.2.1, in this section we will require the following families of sets: for each $m \in \mathbb{N}$ , we set

\begin{equation*} \mathscr{G}_{m} \,:\!=\, \left\{x\,:\, g(x) \geq m\right\}, \qquad \mathscr{H}_{m} \,:\!=\, \left\{x\,:\, h(x) \geq m \right\}, \quad \text{ and } \quad \mathscr{M}_{m} \,:\!=\, \mathscr{G}_m \cup \mathscr{H}_m.\end{equation*}

Proof. This is similar to the proof of Theorem 3.1; however, we require some different notation. For each $m \in \mathbb{N}$ , let ${\mathcal{T}}^{m} = ({\mathcal{T}}^{m}_{t})_{t \geq 0}$ and ${\mathcal{T}}^{m,m} = ({\mathcal{T}}^{m,m}_{t \geq 0})$ denote analogues of the tree process modified on the sets $\mathscr{G}_{m}$ and $\mathscr{H}_{m}$ . In particular, if we define $g_{m}, h_{m}$ so that

\begin{equation*}g_{m} \,:\!=\, g\mathbf{1}_{\mathscr{G}_{m}^{c}} + m \mathbf{1}_{\mathscr{G}_{m}} \quad \text{ and } \quad h_{m} \,:\!=\, h\mathbf{1}_{\mathscr{H}_{m}^{c}} + m\mathbf{1}_{\mathscr{H}_{m}},\end{equation*}

we define ${\mathcal{T}}^{m}$ with the associated functions $g_{m}, h$ , and ${\mathcal{T}}^{m,m}$ with the associated functions $g_{m}, h_{m}$ . Then, by mimicking the approach from the coupling in Lemma 3.1, for each $m \in \mathbb{N}$ we may couple the processes $(\hat{{\mathcal{T}}}^{m,m}, \hat{{\mathcal{T}}}^{m}, \hat{{\mathcal{T}}})$ so that, for all $t \in \mathbb{N}_0$ , their respective partition functions satisfy ${\mathcal{Z}}^{m,m}_{t} \leq {\mathcal{Z}}^{m}_{t} \leq {\mathcal{Z}}_{t}$ ; for each vertex v ^′ with $W_{v^{\prime}} \in \mathscr{H}_{m}^{c}$ ,

\begin{equation*}{\text{deg}^{+}\big({v^{\prime}, \hat{{\mathcal{T}}}^{m}_t}\big)} \leq {\text{deg}^{+}\big({v, \hat{{\mathcal{T}}}^{m,m}_{t}}\big)};\end{equation*}

and for each vertex v with $W_v \in \mathscr{G}_{m}^{c}$ ,

\begin{equation*}{\text{deg}^{+}({v, \hat{{\mathcal{T}}}_t})} \leq {\text{deg}^{+}\big({v, \hat{{\mathcal{T}}}^{m}_{t}}\big)}.\end{equation*}

In this coupling, at each time-step t, one samples $\hat{{\mathcal{T}}}^{m,m}_t$ first, uses this (with another uniformly distributed random variable) to construct $\hat{{\mathcal{T}}}^{m}_t$ , and then uses this to construct $\hat{{\mathcal{T}}}_{t}$ . Therefore, we have the following claim. For a measurable set B, let $\Xi^{m,m}(t, B)$ and $N^{m,m}_{\geq k}(t,B)$ denote the counterparts of $\Xi(t,B)$ and $N_{\geq k}(t,B)$ with respect to the tree ${\mathcal{T}}^{m,m}$ .

Now note that for all m sufficiently large, ${\mathcal{T}}^{m,m}$ satisfies C1. Indeed, if $\text{ess sup}{(g)} = \infty$ , then because ${{\mathbb{E}\!\left[{h_{m}(W)} \right]}} \leq m$ and $g_{m}$ attains its maximum m on a set of positive measure, this follows from Remark 3.1. Otherwise, for m sufficiently large we have $g_{m} = g$ , and for any $\lambda > \text{ess sup}{(g)}$ ,

\begin{equation*}{{\mathbb{E}\!\left[{\frac{h_{m}(W)}{\lambda - g(W)}} \right]}} < \infty, \quad \text{and, by monotone convergence, } \quad \lim_{m \uparrow \infty} {{\mathbb{E}\!\left[{\frac{h_{m}(W)}{\lambda - g(W)}} \right]}} = \infty.\end{equation*}

Thus, making m larger if necessary, we have that C1 is satisfied for this choice of $\lambda$ . In either case, let $\alpha^{(m)}$ denote the Malthusian parameter associated with ${\mathcal{T}}^{m,m}$ . Then $\alpha^{(m)} > \text{ess sup}{(g_{m})}$ increases as m increases, and even if $\text{ess sup}{(g_{m})} < \infty$ we must have

\begin{equation*} \lim_{m \uparrow \infty} \alpha^{(m)} = \infty.\end{equation*}

Indeed, suppose this were not the case, and $\lim_{m \uparrow \infty} \alpha^{(m)} = \alpha^{\prime} < \infty$ . Then, by monotone convergence,

\begin{equation*}1 = \lim_{m \to \infty} {{\mathbb{E}\!\left[{\frac{h_{m}(W)}{\alpha^{(m)} - g(W)}} \right]}} = {{\mathbb{E}\!\left[{\frac{h(W)}{\alpha^{\prime} - g(W)}}\right]}} = \infty,\end{equation*}

since ${{\mathbb{E}\!\left[{h(W)} \right]}} = \infty$ . Now, the assertions of Theorem 3.2 follow from the claim in a similar manner to the way the assertions of Theorem 3.1 follow from Lemma 3.1.

Now, as in the previous subsection, suppose that g and h are increasing, and ${{\text{Supp} \!\left( {\mu} \right)}} \subseteq [0,{w^{*}}]$ , where ${w^{*}} \,:\!=\, \sup{\left({{\text{Supp} \left( {\mu} \right)}}\right)}$ . The proof of the following corollary is similar to that of Corollary 3.1, and we therefore again leave it to the reader.

Corollary 3.2. Under the above assumptions, with regard to the weak topology,

\begin{equation*}\frac{\Xi(t,\cdot)}{t} \xrightarrow{t \to \infty} \delta_{{w^{*}}}({\cdot}), \quad {almost\ surely.}\end{equation*}

4.1. Main results: convergence in probability of $N_{k}(n, B)/\ell n$ under C2

Theorem 4.1. Assume Condition C2. Then, for any measurable set B, we have

\begin{equation*}\frac{N_{k}(t, B)}{\ell t} \xrightarrow{t \to \infty} {{\mathbb{E}\!\left[{\frac{\alpha}{f(k,W) + \alpha} \prod_{s=0}^{k-1} \frac{f(s,W)}{f(s,W) + \alpha} \mathbf{1}_{B}(W)} \right]}} = p^{\alpha}_{k}(B), \quad {in\ probability.}\end{equation*}

In order to prove Theorem 4.1, we define the following family of sets:

(28) \begin{equation} \mathscr{F} \,:\!=\, \left\{B \,:\, B \text{ is measurable and } \forall s \in \mathbb{N}_0, \; f(s,w) \text{ is bounded for $w \in B$} \right\}.\end{equation}

We also require Proposition 4.1 and Proposition 4.2, proved in Section 4.4.1 and Section 4.5.1. These proofs rely on the results stated in Section 4.2 and Section 4.3.

Proposition 4.1. For any set $B\in {\mathscr{F}}$ , for each $k \in \mathbb{N}_0$ we have

\begin{equation*}\lim_{t \to \infty}\frac{{{\mathbb{E}\!\left[{N_{k}(t,B)} \right]}}}{\ell t} = p^{\alpha}_{k}(B).\end{equation*}

Proposition 4.2. For any $B \in {\mathscr{F}}$ and $k \in \mathbb{N}_0$ we have

\begin{equation*}\lim_{t \to \infty} {{\mathbb{E}\!\left[{\frac{\left(N_{k}(t, B)\right)^2}{\ell^2 t^2}}\right]}} = (p^{\alpha}_{k}(B))^2.\end{equation*}

Proof of Theorem 4.1. The result follows for all $B \in {\mathscr{F}}$ by combining Proposition 4.1 and Proposition 4.2 and applying Chebyshev’s inequality.

Now, let B be an arbitrary measurable set and let $\varepsilon > 0$ be given. Then, since for each $s \in \left\{1, \ldots, k\right\}$ the map $w \mapsto f(s,w)$ is measurable, by Lusin’s theorem, we can find a compact set $E \subseteq B$ such that $\mu(B \cap E^{c}) < \varepsilon/3$ and for each $s \in \left\{1, \ldots, k\right\}$ the restriction of the map $w \mapsto f(s,w)$ to E is continuous. Moreover, note that $p^{\alpha}_{k}(B) - p^{\alpha}_{k}(B \cap E) \leq \mu(B \cap E^{c}) < \varepsilon/3$ . Then,

(29) \begin{align} \nonumber {{\mathbb{P}\!\left({\left|\frac{N_{k}(t,B)}{\ell t} - p^{\alpha}_{k}(B)\right| > \varepsilon}\right)}}& \leq \mathbb P \bigg(\bigg(\left|\frac{N_{k}(t,B)}{\ell t} - \frac{N_{k}(t,B\cap E)}{\ell t}\right|\\ \nonumber& \qquad \ + \left|\frac{N_{k}(t,B\cap E)}{\ell t} - p^{\alpha}_{k}(B\cap E)\right| \\ \nonumber & \qquad \ + \left|p^{\alpha}_{k}(B \cap E) - p^{\alpha}_{k}(B)\right|\bigg) > \varepsilon\bigg) \\ \nonumber & \leq {{\mathbb{P}\!\left({\left|\frac{N_{k}(t,B\cap E)}{\ell t} - p^{\alpha}_{k}(B\cap E)\right| > \varepsilon/3}\right)}}\\ & \quad + {{\mathbb{P}\!\left({\left|\frac{N_{k}(t, B)}{\ell t} - \frac{N_{k}(t, B \cap E)}{\ell t}\right| > \varepsilon/3}\right)}}.\end{align}

Now, note that by the strong law of large numbers applied to $N_{\geq 0}(t,B \cap E^{c})/\ell t$ , i.e., the proportion of all vertices with weight belonging to $B \cap E^{c}$ , and Egorov’s theorem, for any $\delta > 0$ there exists an event G with $\mathbb{P}(G) < \delta$ such that

\begin{equation*}\limsup_{t \to \infty} \left(\frac{N_{k}(t, B)}{\ell t} - \frac{N_{k}(t, B \cap E)}{\ell t}\right) = \limsup_{t \to \infty} \frac{N_{k}(t, B \cap E^{c})}{\ell t} \leq \mu(B \cap E^{c})\end{equation*}

uniformly on the complement of G. Therefore, the result follows from (29), Proposition 4.1, and Proposition 4.2 by taking limits as t tends to infinity.

Using the approach to the upper bound for the mean in the next subsection, and applying Corollary 4.1 stated below with $k=1$ and $e_0, e_1 = 0$ , if $N_{\geq 1}(t, B)$ denotes the number of vertices of out-degree at least 1 in the tree with weight belonging to B, we actually have

\begin{equation*}\limsup_{t \to \infty} \frac{{{\mathbb{E}\!\left[{N_{\geq 1}(t, B)} \right]}}}{\ell t} \leq \frac{1}{\alpha^{\prime}}{{\mathbb{E} \!\left[{f(0, W) \mathbf{1}_{B}(W)} \right]}},\end{equation*}

as long as $\liminf_{t \to \infty} \frac{{\mathcal{Z}}_{t}}{t} \geq \alpha^{\prime}$ . By sending $\alpha^{\prime}$ to infinity, this yields the following analogue of Theorem 3.2.

Theorem 4.2. Suppose ${\mathcal{T}}$ is a ${\left({\mu, f, \ell}\right)}$ -RIF tree such that $\lim_{t \to \infty} \frac{{\mathcal{Z}}_{t}}{t} = \infty$ . Then for any measurable set $B \subseteq [0,\infty)$ , we have

\begin{equation*}\frac{N_0(t, B)}{\ell t} \xrightarrow{t\to \infty} \mu(B), \quad {in\ probability}.\end{equation*}

4.2. Summation arguments

Here we state some summation arguments required for the subsequent proofs. The following lemma and corollary are taken from [Reference Fountoulakis, Iyer, Mailler and Sulzbach17]. We include them here, with minor changes in notation, for completeness. For $e_0, \ldots, e_k \geq 0$ , $0 \leq \eta < 1$ , let

\begin{equation*}\mathcal{S}_t(e_0, \ldots, e_k, \eta) \,:\!=\, \frac{1}{t} \sum_{\eta t < i_0 < \cdots < i_k \leq t} \prod_{s=0}^{k-1}\left( \left(\frac{i_s}{i_{s+1}} \right)^{e_s} \cdot \frac{1}{i_{s+1} -1} \right) \left(\frac{i_k}{t} \right)^{e_k}.\end{equation*}

Lemma 4.1. ([Reference Fountoulakis, Iyer, Mailler and Sulzbach17, Lemma 4].) Uniformly in $e_0, \ldots, e_k \geq 0$ , $0 \leq \eta \leq 1/2$ , we have

\begin{equation*}\mathcal{S}_t(e_0, \ldots, e_k, \eta) = \prod_{s=0}^{k} \frac{1}{e_s + 1} + \theta(\eta) + O \left(\frac{1}{t^{1/(k+2)}} + \frac{\sum_{s =0}^k e_s \log^{k+1}(t) }{t} \right).\end{equation*}

Here, $\theta(\eta)$ is a term satisfying $ |\theta(\eta)| \leq M\eta^{1/(k+2)}$ for some universal constant M depending only on k.

Corollary 4.1. ([Reference Fountoulakis, Iyer, Mailler and Sulzbach17, Corollary 5].) For $e_0, \ldots, e_k, f_0, \ldots, f_{k-1} \geq 0$ , $0 \leq \eta \leq 1/2$ , we have

\begin{align*}&\frac 1 t \sum_{\eta t < i_0\leq t} \sum_{{\mathcal{I}}_k \in \tiny\left(\begin{array}{c}\left\{i_0 +1, \ldots, t\right\}\\k\end{array}\right)} \prod_{s=0}^{k-1}\left( \left(\frac{i_s}{i_{s+1}} \right)^{e_s} \cdot \frac{f_{s}}{i_{s+1} -1} \right) \left(\frac{i_k}{t} \right)^{e_k}\\& \hspace {2cm} = \frac{1}{e_k +1}\prod_{s=0}^{k-1} \frac{f_{s}}{e_{s}+1} + \theta^{\prime}(\eta) + O \left(\frac{1}{t^{1/(k+2)}}\right).\end{align*}

Here, $\theta^{\prime}(\eta)$ is a term satisfying $ |\theta^{\prime}(\eta)| \leq M^{\prime}\eta^{1/(k+2)}$ for some universal constant M^′ depending only on k and $f_0, \ldots, f_{k-1}$ , and the constant in the big-O term may depend on $e_0, \ldots, e_{k}, f_0, \ldots, f_k$ .

4.3. Upper bound for the mean of $N_{k}(t,B)/\ell t$

In the following subsections, unless otherwise specified, we let B denote an arbitrary element of the family ${\mathscr{F}}$ defined in (4.1). Let $N_{\eta, k}(t,B)$ be the number of vertices of degree $k \ell$ with weight in B that arrived after time $\eta t$ . Then, since $N_{\eta, k}(t,B) \leq N_{k}(t,B) \leq N_{\eta, k}(t,B) + \eta \ell t$ , we have

(30) \begin{equation} {{\mathbb{E}\!\left[{\left|\frac{N_{\eta, k}(t, B)}{\ell t} - \frac{N_{k}(t, B)}{\ell t}\right|} \right]}} \leq \eta.\end{equation}

Thus, to obtain an upper bound for the convergence of the mean, it suffices to prove that

\begin{equation*}\limsup_{\eta \to 0} \limsup_{t \to \infty} {{\mathbb{E}\!\left[{\frac{N_{\eta, k}(t, B)}{\ell t}} \right]}} = p^{\alpha}_{k}(B).\end{equation*}

In what follows, we use the notation $d_{i}(t)$ to denote the out-degree at time t of the vertex i born at time $i_0 \,:\!=\, \lfloor i/\ell \rfloor$ . We then have

\begin{equation*}{{\mathbb{E}\!\left[{N_{\eta, k}(t,B)} \right]}} = \sum_{\eta t < i_{0} \leq t - k} \ell \cdot {{\mathbb{P}\!\left({d_{i}(t) = k, W_i \in B} \right)}}, \end{equation*}

since the probability is identical for each of the $\ell$ vertices born at each time $i_0$ . In what follows, for a given i we denote by $\mathcal{I}_{k} \,:\!=\, \{i_1, \ldots, i_k\}$ a collection of natural numbers $i_0 < i_1 < \ldots < i_k \leq t$ . For ease of notation we exclude the dependence of $\mathcal{I}_{k}$ on i.

For a natural number $s > i_0$ , we use the notation $i {\rightarrow} s$ to denote that i is the vertex chosen at the sth time-step; hence i gains $\ell$ new neighbours at time s. Likewise, the notation $i {\not \rightarrow} s$ denotes that i is not chosen at the sth time-step. Then, let ${\mathcal{E}}_{i}({\mathcal{I}}_k, B)$ denote the event that $W_i \in B$ and for all $s \in \left\{i_0 +1, \ldots, t\right\}$ , $i {\rightarrow} s$ if and only if $s \in {\mathcal{I}}_k$ . Clearly, we have

\begin{equation*}{{\mathbb{P}\!\left({d_{i}(t) = k, W_i \in B}\right)}} = \sum_{{\mathcal{I}}_k \in \tiny\left(\begin{array}{c}\left\{i_0 +1, \ldots, t\right\}\\k\end{array}\right)} {{\mathbb{P}\!\left({{\mathcal{E}}_{i}({\mathcal{I}}_k, B)}\right)}},\end{equation*}

where $\tiny\left(\begin{array}{c}\left\{i_0 +1, \ldots, t\right\}\\k\end{array}\right)$ denotes the set of all subsets of $\left\{i_0 +1, \ldots, t\right\}$ of size k. For $\varepsilon > 0$ and $t \geq 0$ and natural numbers $N_1 \leq N_2$ , we let

(31) \begin{equation} \mathcal G_\varepsilon (t)= \left\{ \left| \mathcal{Z}_t- \alpha t \right| < \varepsilon \alpha t \right\}, \text{ and } \mathcal G_\varepsilon(N_1, N_2) = \bigcap_{t=N_1}^{N_2}\mathcal G_\varepsilon(t).\end{equation}

Moreover, for $t \geq 1$ , we denote by $\mathscr T_{t}$ the $\sigma$ -field generated by $({\mathcal{T}}_s)_{1 \leq s \leq t}$ , containing all the information generated by the process up to time t. By the assumption of almost sure convergence and Egorov’s theorem, for any $\delta, \varepsilon > 0$ , there exists $N^{\prime} = N^{\prime}(\varepsilon, \delta)$ such that, for all $t \geq N^{\prime}$ , ${{\mathbb{P}\!\left({\mathcal G_\varepsilon(N^{\prime}, t)}\right)}} \geq 1- \delta$ . Thus, for $t \geq N^{\prime}/\eta$ , we have

(32) \begin{align} {{\mathbb{E}\!\left[{N_{\eta, k}(t,B)} \right]}} &\leq {{\mathbb{E}\!\left[{N_{\eta, k}(t,B) \mathbf{1}_{\mathcal{G}_{\varepsilon}(N^{\prime},t)}} \right]}} + \ell t\!\left(1-{{\mathbb{P}\!\left({\mathcal G_\varepsilon(N^{\prime}, t)}\right)}} \right)\\& \leq \ell \left(\sum_{\eta t < i_0 \leq t} \sum_{{\mathcal{I}}_k \in \tiny\left(\begin{array}{c}\left\{i_0 +1, \ldots, t\right\}\\k\end{array}\right)} {{\mathbb{P}\left({{\mathcal{E}}_{i}({\mathcal{I}}_k, B) \cap \mathcal{G}_{\varepsilon}(i_0,t)}\right)}} + \delta t \right).\nonumber \end{align}

We use the shorthand $\alpha_{\pm \varepsilon} \,:\!=\, (1 \pm \varepsilon) \alpha$ .

Proposition 4.3. Let $B \in {\mathscr{F}}$ and $0 < \varepsilon, \eta \leq 1/2$ . As $t \to \infty$ , uniformly in $\eta t < i_0 \leq t -k$ , ${\mathcal{I}}_k = \{i_1, \ldots, i_k\}\in \tiny\left(\begin{array}{c}\left\{i_0 +1, \ldots, t\right\}\\k\end{array}\right)$ , and the choice of $\varepsilon$ , we have

\begin{align*}& {{\mathbb{P}\!\left({{\mathcal{E}}_{i}({\mathcal{I}}_k, B) \cap \mathcal{G}_{\varepsilon}(i_0,t)}\right)}}\\ & \leq \left(1 + O(1/t) \right) {{\mathbb{E}\!\left[{\left(\frac{i_{k}}{t}\right)^{f(k,W)/\alpha_{+\varepsilon}} \prod_{s=0}^{k-1} \left( \frac{i_{s}}{i_{s +1}}\right)^{f(s, W)/\alpha_{+\varepsilon}} \frac{f(s,W)}{\alpha_{-\varepsilon}(i_{s+1} - 1)} \mathbf{1}_{B}(W)} \right]}}.\end{align*}

Corollary 4.2. Let $B \in {\mathscr{F}}$ and $0 < \delta, \varepsilon, \eta \leq 1/2$ . Then there exists $N = N(\delta, \varepsilon, \eta)$ such that, for all $t \geq N$ ,

\begin{align*}& \frac{{{\mathbb{E}\!\left[{N_{\eta, k}(t,B)} \right]}}}{\ell t} \leq (1+\delta) \left(\frac{1+\varepsilon}{1-\varepsilon}\right)^k {{\mathbb{E}\!\left[{ \frac{\alpha_{+\varepsilon}}{f(k, W)+\alpha_{+\varepsilon}} \prod_{s=0}^{k-1} \frac{f(s, W)}{f(s, W)+\alpha_{+\varepsilon}} \mathbf{1}_{B}(W)} \right]}}\nonumber\\& \qquad\qquad\qquad + C \eta^{1/(k+2)} + \delta,\end{align*}

where the constant C may depend on k and B but not on n and not on the choices of $\delta, \varepsilon, \eta$ . In particular, for each $B \in \mathscr{F}$ and $k \in \mathbb{N}_0$ ,

\begin{equation*}\limsup_{t \to \infty} {{\mathbb{E}\!\left[{N_k(t,B)} \right]}}/ \ell t \leq p^{\alpha}_k(B).\end{equation*}

We proceed towards the proof of Proposition 4.3. Let $\varepsilon, \eta$ be given such that $0 < \varepsilon, \eta \leq 1/2$ . For $\eta t < i_0 \leq t $ and ${\mathcal{I}}_k = \{i_1, \ldots, i_k\} \in \tiny\left(\begin{array}{c}\left\{i_0 +1, \ldots, t\right\}\\k\end{array}\right)$ , for each $s \in \left\{i_0 +1, \ldots, t\right\}$ we define

\begin{equation*} {\mathcal{D}}_{s} \,:\!=\,\begin{cases}\left\{i{\rightarrow} s\right\} & \text{ if } s\in{\mathcal{I}}_k,\\\left\{i{\not \rightarrow} s\right\} & \text{ otherwise},\end{cases} \end{equation*}

and $\tilde {\mathcal{D}}_s = {\mathcal{D}}_s \cap \mathcal G_{\varepsilon}(s).$ We also define $\tilde {\mathcal{D}}_{i_0} = \mathcal G_\varepsilon(i_0) \cap \{W_i \in B\}$ , and for simplicity of notation we write $D_j$ and $\tilde D_j$ for the indicator random variables $\mathbf{1}_{{\mathcal{D}}_j}$ and $\mathbf{1}_{\tilde {\mathcal{D}}_j}$ , respectively. Note that $\mathcal E_i({\mathcal{I}}_k, B) \cap \mathcal G_\varepsilon(i_0,t) = \bigcap_{j=i_0}^t \tilde {\mathcal{D}}_j$ . To bound the probability of this event, we define

\begin{equation*} X_s = {{\mathbb{E}\!\left[{\prod_{j=i_{s}+1}^{t} \tilde D_j \; \bigg | \; \mathscr T_{i_s}} \right]}} \tilde D_{i_s},\quad s \in \left\{0, \ldots, k\right\}\end{equation*}

and observe that ${{\mathbb{E}\!\left[{X_0} \right]}} = {{\mathbb{P}\!\left({\bigcap_{s=i_0}^t \tilde {\mathcal{D}}_s}\right)}}$ is the probability we seek.

Lemma 4.2. For $s \in \left\{0, \ldots, k\right\}$ , we have

(33) \begin{equation} X_s \leq \prod_{u=i_{k}+1}^{n} \left(1 - \frac{f(k,W)}{\alpha_{+\varepsilon}(u- 1)}\right) \left(\prod_{j = s}^{k-1} \frac{f(j,W)}{\alpha_{-\varepsilon}(i_{j+1}-1)}\prod_{j^{\prime}= i_j +1}^{i_{j+1} -1} \left(1 - \frac{f(j,W)}{\alpha_{+\varepsilon}(j^{\prime} -1)}\right)\right)\tilde{D}_{i_s},\end{equation}

where we interpret any empty products (for example when $i_k = n$ ) as equal to 1. In particular,

(34) \begin{align}{{\mathbb{E}\!\left[{X_0} \right]}} &\leq {{\mathbb{E}\!\left[{\prod_{u=i_{k}+1}^{n}\!\! \left(1 - \frac{f(k,W)}{\alpha_{+\varepsilon}(u- 1)}\right) \left(\prod_{j = 0}^{k-1} \frac{f(j,W)}{\alpha_{-\varepsilon}(i_{j+1}-1)} \prod_{j^{\prime}= i_j +1}^{i_{j+1} -1} \!\left(1 - \frac{f(j,W)}{\alpha_{+\varepsilon}(j^{\prime} -1)}\right)\!\right) \mathbf{1}_{B}(W)} \right]}}. \qquad\end{align}

Proof. We prove (33) by backwards induction. For the base case, $s=k$ , if $i_k = n$ , the inequality is trivial, as $X_k = \tilde{D}_{i_k}$ . Thus, assuming $i_k < n$ , by the tower property,

\begin{align*}{{\mathbb{E}\!\left[{\prod_{j=i_{k}+1}^{n} \tilde D_j \; \bigg | \; \mathscr T_{i_k}} \right]}} & = {{\mathbb{E} \!\left[{{{\mathbb{E}\!\left[{\tilde D_{n} \; \bigg | \; \mathscr{T}_{n-1}} \right]}}\prod_{j=i_k+1}^{n-1} \tilde D_j \; \bigg | \; \mathscr T_{i_k}} \right]}} \\ &\leq {{\mathbb{E}\!\left[{{{\mathbb{E}\!\left[{D_{n} \; \bigg | \; \mathscr{T}_{n-1}} \right]}}\prod_{j=i_k+1}^{n-1} \tilde D_j \; \bigg | \; \mathscr T_{i_k}} \right]}} \\ &= {{\mathbb{E}\!\left[{\left(1 - \frac{f(k,W)}{\mathcal{Z}_{n-1}}\right)\prod_{j=i_k+1}^{n-1} \tilde D_j \; \bigg | \; \mathscr T_{i_k}} \right]}} \\ &\leq \left(1 - \frac{f(k,W)}{ \alpha_{+\varepsilon}(n-1)}\right) {{\mathbb{E}\!\left[{\prod_{j=i_k+1}^{n-1} \tilde D_j \; \bigg | \; \mathscr T_{i_k}} \right]}},\end{align*}

and iterating this argument with the conditional expectation on the right-hand side proves the base case. Now, note that for $s \in \left\{0, \ldots, k-1\right\}$

\begin{equation*}X_s = {{\mathbb{E}\!\left[{X_{s+1} \prod_{j=i_{s} +1}^{i_{s+1} -1} \tilde D_{j} \; \bigg | \; \mathscr T_{i_s}} \right]}} \tilde D_{i_s}.\end{equation*}

Applying the induction hypothesis, it suffices to bound the term ${{\mathbb{E}\!\left[{\prod_{j=i_{s} +1}^{i_{s+1}} \tilde D_{j} \; \bigg | \; \mathscr T_{i_s}}\right]}}$ , and, similarly to the base case, we may assume $i_{s} < i_{s+1} - 1$ . But then we have

\begin{align*}{{\mathbb{E}\!\left[{\prod_{j=i_{s} +1}^{i_{s+1}} \tilde D_{j} \; \bigg | \; \mathscr T_{i_s}} \right]}} & = {{\mathbb{E} \!\left[{{{\mathbb{E}\!\left[{\tilde D_{i_{s+1}} \; \bigg | \; \mathscr{T}_{i_{s+1} -1}} \right]}} \prod_{j=i_{s} +1}^{i_{s+1}-2} \tilde D_{j} \; \bigg | \; \mathscr T_{i_s}}\right]}} \\ &\leq {{\mathbb{E}\!\left[{{{\mathbb{E}\!\left[{D_{i_{s+1}} \; \bigg | \; \mathscr{T}_{i_{s+1} -1}}\right]}} \prod_{j=i_{s} +1}^{i_{s+1}-2} \tilde D_{j} \; \bigg | \; \mathscr T_{i_s}}\right]}}\\ &\leq \frac{f(s,W)}{\alpha_{-\varepsilon}(i_{s+1} - 1)}{{\mathbb{E}\!\left[{\prod_{j=i_s + 1}^{i_{s+1}-2} \tilde{D}_{j} \; \bigg | \; \mathscr T_{i_s}}\right]}}\\ & \leq \frac{f(s,W)}{\alpha_{-\varepsilon}(i_{s+1} - 1)}{{\mathbb{E}\!\left[{{{\mathbb{E}\!\left[{D_{i_{s+1} -1} \; \bigg | \; \mathscr{T}_{i_{s+1} -1}}\right]}} \prod_{j=i_{s} +1}^{i_{s+1}-2} \tilde D_{j} \; \bigg | \; \mathscr T_{i_s}}\right]}}\\ & \leq \frac{f(s,W)}{\alpha_{-\varepsilon}(i_{s+1} - 1)}\left(1 - \frac{f(s,W)}{\alpha_{+\varepsilon} \left(i_{s+1} -2\right) } \right) {{\mathbb{E}\!\left[{ \prod_{j=i_{s} +1}^{i_{s+1}-2} \tilde D_{j} \; \bigg | \; \mathscr T_{i_s}}\right]}}.\end{align*}

Iterating these bounds, the inductive step follows in a similar manner to the base case. Finally, noting that $\mathbf{1}_{\tilde{\mathcal{D}}_{i}} \leq \mathbf{1}_{B}(W)$ proves (34).

The next lemma follows from a simple application of Stirling’s formula, i.e., (25).

Lemma 4.3. Let $\eta, C > 0$ . Then, uniformly over $\eta t \leq a \leq b$ and $0 \leq \beta \leq C$ , we have

\begin{equation*}\prod_{j=a+1}^{b-1} \left(1-\frac{\beta}{j-1}\right) = \left(\frac{a}{b}\right)^\beta \left( 1 + O\!\left(\frac 1 t\right) \right).\end{equation*}

4.4. Deducing convergence of the mean of $N_{k}(t,B)/\ell t$

In this subsection we deduce a lower bound on $\liminf_{t \to \infty} {{\mathbb{E}\!\left[{N_k(t,B)}\right]}}/ \ell t$ on measurable sets $B \in {\mathscr{F}}$ . In what follows, denote by $N_{\geq M}(t,B)$ the number of vertices of out-degree $\geq \ell M$ with weight belonging to B. Moreover, let $N(t,B) = N_{\geq 0}(t,B)$ denote the total number of vertices at time t with weight belonging to B.

Lemma 4.4. For any measurable set B, we have

\begin{equation*}\limsup_{t \to \infty} \frac{N_{\geq M}(t,B)}{\ell t} \leq \frac{1}{M}\end{equation*}

almost surely.

4.4.1. Proof of Proposition 4.1

Proof. Recall that Corollary 4.2 showed that for each $B \in \mathscr{F}$ and $k \in \mathbb{N}_0$ ,

\begin{equation*}\limsup_{t \to \infty} {{\mathbb{E}\!\left[{N_k(t,B)}\right]}}/ \ell t \leq p^{\alpha}_k(B).\end{equation*}

Now, suppose that Proposition 4.1 fails, so that in particular there exist some set $B^{\prime} \in \mathscr{F}$ and some $k^{\prime} \in \mathbb{N}_{0}$ such that

\begin{equation*} \liminf_{t \to \infty} \frac{{{\mathbb{E}\!\left[{N_{k^{\prime}}(t,B^{\prime})}\right]}}}{\ell t} < p^{\alpha}_{k^{\prime}}(B^{\prime}).\end{equation*}

Thus, for some $\epsilon^{\prime} > 0$ , we have

\begin{equation*}\liminf_{t \to \infty} \frac{{{\mathbb{E}\!\left[{N_{k^{\prime}}(t,B^{\prime})}\right]}}}{\ell t} \leq p^{\alpha}_{k^{\prime}}(B) - \epsilon^{\prime}.\end{equation*}

Now, using Lemma 4.4, choose $M > \max{\left \{k^{\prime}, \frac{2}{\epsilon^{\prime}}\right \}}$ , so that

\begin{equation*}\limsup_{t \to \infty} \frac{N_{\geq M}(t,B^{\prime})}{\ell t} < \epsilon^{\prime}/2.\end{equation*}

Then, recalling (13),

(35) \begin{align}\liminf_{t \to \infty} {{\mathbb{E}\!\left[{\sum_{k=0}^{M} \frac{N_{k}(t,B^{\prime})}{\ell t}}\right]}} & \leq \liminf_{t \to \infty} {{\mathbb{E}\!\left[{\frac{N_{k^{\prime}}(t,B^{\prime})}{\ell t}} \right]}} + \sum_{k \neq k^{\prime}} \limsup_{t \to \infty} {{\mathbb{E} \!\left[{\frac{N_{k}(t,B^{\prime})}{\ell t}} \right]}}\\ & \leq \left(\sum_{k=0}^{\infty} p^{\alpha}_{k}(B^{\prime})\right) - \epsilon^{\prime} \leq \mu(B^{\prime}) - \epsilon^{\prime}.\nonumber\end{align}

On the other hand, by Fatou’s lemma, we have

(36) \begin{align}\liminf_{t \to \infty} {{\mathbb{E}\!\left[{\sum_{k=0}^{M} \frac{N_{k}(t,B^{\prime})}{\ell t}}\right]}} & \geq {{\mathbb{E} \!\left[{\liminf_{t \to \infty} \sum_{k=0}^{M} \frac{N_{k}(t,B^{\prime})}{\ell t}} \right]}} \end{align}

\begin{align} \nonumber &= {{\mathbb{E}\!\left[{\liminf_{t \to \infty}\left(\frac{N(t,B^{\prime})}{\ell t} - \frac{N_{\geq M}(t,B^{\prime})}{\ell t}\right)} \right]}}\geq \mu(B^{\prime}) - \epsilon^{\prime}/2,\end{align}

where the last inequality follows from the strong law of large numbers. But then, combining (35) and (36), we have $\mu(B^{\prime}) - \epsilon^{\prime} \geq \mu(B^{\prime}) - \epsilon^{\prime}/2$ , a contradiction.

4.5. Second moment calculations

In order to bound the second moment, we apply calculations similar to those at the start of the section to compute asymptotically the number of pairs of vertices of out-degree $k \ell$ born after time $\eta t$ . For vertices i and j, as in Section 4.3, we set $i_0 \,:\!=\, \lfloor i/\ell \rfloor$ and $j_0 \,:\!=\, \lfloor j/\ell \rfloor$ , and note that

(37) \begin{align} {{\mathbb{E}\!\left[{\left(N_{\eta, k}(t,B)\right)^2}\right]}} = \sum_{\eta t < i_0, j_0 \leq t-k} \sum_{j\,:\, \lfloor j/\ell \rfloor = j_0} \sum_{i\,:\, \lfloor i/\ell \rfloor = i_0 } {{\mathbb{P}\!\left({d_{i}(t) = k, W_i \in B, d_{j}(t) = k, W_j \in B}\right)}}.\end{align}

Note that, in a similar manner to (30), we have

\begin{equation*}{{\mathbb{E}\!\left[{\left|\frac{\left(N_{\eta, k}(t,B)\right)^2}{\ell^2 t^2} - \frac{\left(N_{k}(t,B)\right)^2}{\ell^2 t^2}\right|}\right]}} \leq \eta, \end{equation*}

so that it suffices to prove that

\begin{equation*}\limsup_{\eta \to 0} \limsup_{t \to \infty} {{\mathbb{E}\!\left[{\frac{\left(N_{\eta, k}(t, B)\right)^2}{\ell^2 t^2}}\right]}} \leq (p^{\alpha}_{k}(B))^2.\end{equation*}

Recall that, for a given i, we denote by ${\mathcal{I}}_k$ a collection of natural numbers $i_0 < i_1 < \cdots < i_k \leq t$ . Moreover, for a given j, we denote by ${\mathcal{J}}_{k}$ a collection of natural numbers $j_0 < j_1 < \cdots < j_k \leq t$ . Similarly to Section 4.3, for $s > j$ we use the notation $j {\rightarrow} s$ to denote that j is the vertex chosen at the sth time-step, and likewise, we let ${\mathcal{E}}_{j}({\mathcal{J}}_k, B)$ denote the event that $W_j \in B$ and for all $s \in \left\{j_0 +1, \ldots, t\right\}$ , $j {\rightarrow} s$ if and only if $s \in {\mathcal{J}}_k$ . Then we have

\begin{align*}&\mathbb{P}\left({d_{i}(t) = k, W_i \in B, d_{j}(t) = k, W_j \in B}\right)\\& \qquad \qquad \qquad = \sum_{{\mathcal{J}}_k \in \tiny\left(\begin{array}{c}\left\{j_0 +1, \ldots, t\right\}\\k\end{array}\right)} \sum_{{\mathcal{I}}_k \in \tiny\left(\begin{array}{c}\left\{i_0 +1, \ldots, t\right\}\\k\end{array}\right)} {{\mathbb{P}\!\left({{\mathcal{E}}_i({\mathcal{I}}_k,B) \cap {\mathcal{E}}_j({\mathcal{J}}_k, B)}\right)}}.\end{align*}

Note that the contribution to the above sum corresponding to terms with ${\mathcal{I}}_{k} \cap {\mathcal{J}}_{k} \neq \varnothing$ , and $i \neq j$ , is zero, since it is impossible for distinct vertices to be chosen in a single time-step. But then, the terms corresponding to $i = j$ contribute at most ${{\mathbb{E}\!\left[{N_{\eta,k}(n, B)}\right]}} \leq \ell n$ to the right side of (37). Next, for any choice of indices with ${{\mathcal{I}}_{k} \cap {\mathcal{J}}_{k} = \emptyset}$ , there are at most $\ell^2$ pairs of vertices (i, j) born at respective times $(i_0, j_0)$ contributing to the sum in (37). Recalling the definitions of $\mathcal{G}_{\varepsilon}(t)$ , $\mathcal{G}_{\varepsilon}(N_1,N_2)$ , and $N^{\prime} = N^{\prime}(\varepsilon, \delta)$ from (31) and below in the previous subsection, in a similar manner to (32) we have, for $t \geq N^{\prime}/\eta$ ,

(38) \begin{equation} {{\mathbb{E}\!\left[{\left(N_{\eta, k}(t, B) \right)^2}\right]}} \leq \ell^2 \left( \sum_{\eta t < i_0, j_0 \leq t-k} \sum_{{{\mathcal{I}}_{k} \cap {\mathcal{J}}_{k} = \emptyset}} {{\mathbb{P}\!\left({{\mathcal{E}}_i({\mathcal{I}}_k,B) \cap {\mathcal{E}}_j({\mathcal{J}}_k, B) \cap \mathcal{G}_{\varepsilon}(i_0, t)}\right)}} + \delta t^2\right) + \ell t.\end{equation}

We then have the following.

Proposition 4.4. Let $B \in {\mathscr{F}}$ and $0 < \varepsilon, \eta \leq 1/2$ . As $t \to \infty$ , uniformly in $\eta t < i_0 \leq j_0 \leq t -k$ , in ${\mathcal{I}}_k \in \tiny\left(\begin{array}{c}\left\{i_0 +1, \ldots, t\right\}\\k\end{array}\right)$ , ${\mathcal{J}}_k \in \tiny\left(\begin{array}{c}\left\{j_0+1, \ldots, t\right\}\\k\end{array}\right)$ such that ${{\mathcal{I}}_{k} \cap {\mathcal{J}}_{k} = \emptyset}$ , and in the choice of $\varepsilon$ , we have

(39) \begin{align} & \nonumber{{\mathbb{P}\!\left({{\mathcal{E}}_i({\mathcal{I}}_k,B) \cap {\mathcal{E}}_j({\mathcal{J}}_k, B) \cap \mathcal{G}_{\varepsilon}(i_0, t)}\right)}} \nonumber\\ & \qquad\qquad \leq (1 + O(1/t)) {{\mathbb{E}\!\left[{\left(\frac{i_{k}}{t}\right)^{f(k,W)/\alpha_{+\varepsilon}} \cdot \prod_{s=0}^{k-1} \left(\left( \frac{i_{s}}{i_{s +1}}\right)^{f(s, W)/\alpha_{+\varepsilon}} \frac{f(s,W)}{\alpha_{-\varepsilon}(i_{s+1}-1)}\right) \mathbf{1}_{B}(W)}\right]}}\nonumber\\ & \qquad\qquad \times {{\mathbb{E}\!\left[{\left(\frac{j_{k}}{t}\right)^{f(k,W)/\alpha_{+\varepsilon}} \cdot \prod_{s=0}^{k-1} \left(\left( \frac{j_{s}}{j_{s +1}}\right)^{f(s, W)/\alpha_{+\varepsilon}} \frac{f(s,W)}{\alpha_{-\varepsilon}(j_{s+1}-1}\right)\mathbf{1}_{B}(W)}\right]}}.\end{align}

We leave the details of the proof of this proposition to the reader, as it follows an analogous approach to the proof of Proposition 4.3, using a backwards induction argument.

Proof. sketch. Let $u_1, \ldots, u_{2k}$ denote the indices in ${\mathcal{I}}_k \cup {\mathcal{J}}_k$ , and let $f_x(i), f_{x}(j)$ denote the fitnesses associated with vertex i and vertex j at time x. Then, when we bound the probabilities $\{i {\not \rightarrow} x \} \cap \{j {\not\rightarrow} x\}$ for all $x \in \left\{u_{s} +1, \ldots, u_{s+1} - 1\right\}$ from above, we obtain terms of the form

\begin{equation*}\prod_{x = u_{s}+1}^{u_{s+1} - 1} \left(1 - \frac{f_{x}(i) + f_{x}(j)}{\alpha_{+\varepsilon}(x-1)} \right) = \left(\frac{u_{s}}{u_{s+1}} \right)^{f_{x}(i) + f_{x}(j)} \left(1+ O\!\left(\frac{1}{t}\right)\right),\end{equation*}

where the right side follows from Lemma 4.3. Then, when we evaluate the expectation analogous to the expectation appearing in (34), we obtain an expectation involving products of terms dependent on $W_i$ and $W_j$ , i.e., the weights associated with vertex i and vertex j. These terms separate into a product of expectations by the independence of the random variables $W_i$ , $W_j$ , and finally, many of the products telescope to yield the right side of (39).

4.5.1. Proof of Proposition 4.2

Proof. We apply Proposition 4.4 to bound the summands in (38). Then, as we are looking for an upper bound, we may drop the condition ${{\mathcal{I}}_{k} \cap {\mathcal{J}}_{k} = \emptyset}$ when evaluating the sum. But then, by Corollary 4.1, we have, uniformly in $\varepsilon$ and $\eta$ ,

\begin{align*}& \sum_{\eta t < i_0, j_0 \leq t} \sum_{{\mathcal{I}}_k, {\mathcal{J}}_k} {{\mathbb{E} \!\left[{\left(\frac{i_k}{t}\right)^{f(k,W)/\alpha_{+\varepsilon}}\cdot \prod_{s=0}^{k-1} \left(\frac{i_s}{i_{s+1}}\right)^{f(s,W)/\alpha_{+\varepsilon}}\frac{f(s,W)}{\alpha_{-\varepsilon} (i_{s+1}-1)} \mathbf{1}_{B}(W)} \right]}} \nonumber \nonumber \\ &\qquad\qquad\qquad \times {{\mathbb{E}\!\left[{\left(\frac{j_k}{t}\right)^{f(k,W)/\alpha_{+\varepsilon}} \cdot \prod_{s=0}^{k-1} \left(\frac{j_s}{j_{s+1}}\right)^{f(s,W)/\alpha_{+\varepsilon}}\frac{f(s,W)}{\alpha_{-\varepsilon} (j_{s+1}-1)} \mathbf{1}_{B}(W)}\right]}} \\ & \leq \left(\frac{1+\varepsilon}{1-\varepsilon}\right)^{2k} \left( {{\mathbb{E} \!\left[{\frac{\alpha_{+\varepsilon}}{f(k,W)+\alpha_{+\varepsilon}} \prod_{s=0}^{k-1} \frac{f(s,W)}{f(s,W)+\alpha_{+\varepsilon}} \mathbf{1}_{B}(W)}\right]}} \right)^2+ O\!\left(t^{-1/(k+2)}\right) + C^{\prime}\eta^{1/{k+2}},\end{align*}

for some universal constant $C^{\prime} >0$ , depending only on B, f. The result follows.