We propose a method for cutting down a random recursive tree that focuses on its higher-degree vertices. Enumerate the vertices of a random recursive tree of size n according to the decreasing order of their degrees; namely, let $(v^{(i)})_{i=1}^{n}$ be such that $\deg(v^{(1)}) \geq \cdots \geq \deg (v^{(n)})$. The targeted vertex-cutting process is performed by sequentially removing vertices $v^{(1)}, v^{(2)}, \ldots, v^{(n)}$ and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedure, $K_n$, is upper bounded by $Z_{\geq D}$, which denotes the number of vertices that have degree at least as large as the degree of the root. We prove that $\ln Z_{\geq D}$ grows as $\ln n$ asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the kth moment of $\ln Z_{\geq D}$ is proportional to $(\!\ln n)^k$. As a consequence, we obtain that the first-order growth of $K_n$ is upper bounded by $n^{1-\ln 2}$, which is substantially smaller than the required number of removals if, instead, the vertices were selected uniformly at random.

We present and study a novel algorithm for the computation of 2-Wasserstein population barycenters of absolutely continuous probability measures on Euclidean space. The proposed method can be seen as a stochastic gradient descent procedure in the 2-Wasserstein space, as well as a manifestation of a law of large numbers therein. The algorithm aims to find a Karcher mean or critical point in this setting, and can be implemented ‘online’, sequentially using independent and identically distributed random measures sampled from the population law. We provide natural sufficient conditions for this algorithm to almost surely converge in the Wasserstein space towards the population barycenter, and we introduce a novel, general condition which ensures uniqueness of Karcher means and, moreover, allows us to obtain explicit, parametric convergence rates for the expected optimality gap. We also study the mini-batch version of this algorithm, and discuss examples of families of population laws to which our method and results can be applied. This work expands and deepens ideas and results introduced in an early version of Backhoff-Veraguas et al. (2022), in which a statistical application (and numerical implementation) of this method is developed in the context of Bayesian learning.

The binary contact path process (BCPP) introduced in Griffeath (1983) describes the spread of an epidemic on a graph and is an auxiliary model in the study of improving upper bounds of the critical value of the contact process. In this paper, we are concerned with limit theorems of the occupation time of a normalized version of the BCPP (NBCPP) on a lattice. We first show that the law of large numbers of the occupation time process is driven by the identity function when the dimension of the lattice is at least 3 and the infection rate of the model is sufficiently large conditioned on the initial state of the NBCPP being distributed with a particular invariant distribution. Then we show that the centered occupation time process of the NBCPP converges in finite-dimensional distributions to a Brownian motion when the dimension of the lattice and the infection rate of the model are sufficiently large and the initial state of the NBCPP is distributed with the aforementioned invariant distribution.

We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [6, 19], notably relying on Karamata’s Tauberian Theorem to do so. We use these asymptotics to derive precise large-deviation results in the fashion of [32] for the just-mentioned processes.

We study the noise sensitivity of the minimum spanning tree (MST) of the $n$-vertex complete graph when edges are assigned independent random weights. It is known that when the graph distance is rescaled by $n^{1/3}$ and vertices are given a uniform measure, the MST converges in distribution in the Gromov–Hausdorff–Prokhorov (GHP) topology. We prove that if the weight of each edge is resampled independently with probability $\varepsilon \gg n^{-1/3}$, then the pair of rescaled minimum spanning trees – before and after the noise – converges in distribution to independent random spaces. Conversely, if $\varepsilon \ll n^{-1/3}$, the GHP distance between the rescaled trees goes to $0$ in probability. This implies the noise sensitivity and stability for every property of the MST that corresponds to a continuity set of the random limit. The noise threshold of $n^{-1/3}$ coincides with the critical window of the Erdős-Rényi random graphs. In fact, these results follow from an analog theorem we prove regarding the minimum spanning forest of critical random graphs.

As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure, and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We use martingale theory to prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. We also apply our results to further generalizations of random trees, hooking networks, and continuous spaces constructed from line segments.

We investigate branching processes in varying environment, for which $\overline{f}_n \to 1$ and $\sum_{n=1}^\infty (1-\overline{f}_n)_+ = \infty$, $\sum_{n=1}^\infty (\overline{f}_n - 1)_+ < \infty$, where $\overline{f}_n$ stands for the offspring mean in generation n. Since subcritical regimes dominate, such processes die out almost surely, therefore to obtain a nontrivial limit we consider two scenarios: conditioning on nonextinction, and adding immigration. In both cases we show that the process converges in distribution without normalization to a nondegenerate compound-Poisson limit law. The proofs rely on the shape function technique, worked out by Kersting (2020).

We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in $\{1,\dots,n\}$ occurs k times, where k may depend on n. This generalises the famous Ulam–Hammersley problem of the case $k=1$. The proof relies on poissonisation and on a careful non-asymptotic analysis of variants of the Hammersley–Aldous–Diaconis particle system.

In this paper we extend results on reconstruction of probabilistic supports of independent and identically distributed random variables to supports of dependent stationary ${\mathbb R}^d$-valued random variables. All supports are assumed to be compact of positive reach in Euclidean space. Our main results involve the study of the convergence in the Hausdorff sense of a cloud of stationary dependent random vectors to their common support. A novel topological reconstruction result is stated, and a number of illustrative examples are presented. The example of the Möbius Markov chain on the circle is treated at the end with simulations.

This paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin–Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite-horizon considerations. However, there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin–Wentzell quasipotential is indeed the rate function.

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant, and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of independent and identically distributed random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of independent and identically distributed exponential random variables.

This paper is concerned with stochastic Schrödinger delay lattice systems with both locally Lipschitz drift and diffusion terms. Based on the uniform estimates and the equicontinuity of the segment of the solution in probability, we show the tightness of a family of probability distributions of the solution and its segment process, and hence the existence of invariant measures on $l^2\times L^2((-\rho,\,0);l^2)$ with $\rho >0$. We also establish a large deviation principle for the solutions with small noise by the weak convergence method.

The asymptotic behavior of the Jaccard index in G(n, p), the classical Erdös–Rényi random graph model, is studied as n goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in G(n, p) is shown to be asymptotically normal under an additional mild condition that $np\to\infty$ and $n^2(1-p)\to\infty$.

We consider estimation of the spot volatility in a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices. Based on discrete, noisy observations of an Itô semimartingale with jumps and general stochastic volatility, we present a simple and explicit estimator using local order statistics. We establish consistency and stable central limit theorems as asymptotic properties. The asymptotic analysis builds upon an expansion of tail probabilities for the order statistics based on a generalized arcsine law. In order to use the involved distribution of local order statistics for a bias correction, an efficient numerical algorithm is developed. We demonstrate the finite-sample performance of the estimation in a Monte Carlo simulation.

In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $. We consider a class of non-square-integrable observables $\phi $, mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$, where $x_0$ is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index $\alpha \in (0,2)$. The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $\alpha $, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $\nu $. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

We establish the exponential nonuniform Berry–Esseen bound for the maximum likelihood estimator of unknown drift parameter in an ultraspherical Jacobi process using the change of measure method and precise asymptotic analysis techniques. As applications, the optimal uniform Berry–Esseen bound and optimal Cramér-type moderate deviation for the corresponding maximum likelihood estimator are obtained.

We prove that the local time of random walks conditioned to stay positive converges to the corresponding local time of three-dimensional Bessel processes by proper scaling. Our proof is based on Tanaka’s pathwise construction for conditioned random walks and the derivation of asymptotics for mixed moments of the local time.

We establish the asymptotic expansion in $\beta $ matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a $\frac {1}{N}$ expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ($\beta = 2$) as well as orthogonal ($\beta = 1$) and skew-orthogonal ($\beta = 4$) polynomials outside the bulk.

We consider linear preferential attachment trees with additive fitness, where fitness is the random initial vertex attractiveness. We show that when the fitnesses are independent and identically distributed and have positive bounded support, the local weak limit can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence for the total variation distance between the r-neighbourhoods of a uniformly chosen vertex in the preferential attachment tree and the root vertex of the local weak limit. The proof uses a Pólya urn representation of the model, for which we give new estimates for the beta and product beta variables in its construction. As applications, we obtain limiting results and convergence rates for the degrees of the uniformly chosen vertex and its ancestors, where the latter are the vertices that are on the path between the uniformly chosen vertex and the initial vertex.

We consider Gaussian approximation in a variant of the classical Johnson–Mehl birth–growth model with random growth speed. Seeds appear randomly in $\mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The locations, birth times, and growth speeds of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed, the time distribution, and a weight function $h\;:\;\mathbb{R}^d \times [0,\infty) \to [0,\infty)$, we prove a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Such models have previously been considered, albeit with fixed growth speed. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.