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In this study, we consider a class of multiple-drawing opposite-reinforcing urns with time-dependent replacement rules. The class has the symmetric property of a Friedman-type urn. We divide the class into a small-increment regime and a large-increment regime. For small-increment schemes, we prove almost-sure convergence and a central limit theorem for the proportion of white balls by stochastic approximation. For large-increment schemes, by assuming the affinity condition, we show almost-sure convergence of the proportion of white balls by martingale theory and present a way to identify the limit distribution of the proportion of white balls.
Under the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable Gromov-Hausdorff topology, we establish asymptotic bounds on the distribution of the
$\varepsilon$
-blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the
$\varepsilon$
-blanket times of the random walks if the
$\varepsilon$
-blanket time of the limiting diffusion is continuous at
$\varepsilon$
with probability 1. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical Galton-Watson trees and the Erdős-Rényi random graph in the critical window. We highlight that proving continuity of the
$\varepsilon$
-blanket time of the limiting diffusion relies on the scale invariance of a finite measure that gives rise to realizations of the limiting compact random metric space, and therefore we expect our results to hold for other examples of random graphs with a similar scale invariance property.
One way to model telecommunication networks are static Boolean models. However, dynamics such as node mobility have a significant impact on the performance evaluation of such networks. Consider a Boolean model in $\mathbb {R}^d$ and a random direction movement scheme. Given a fixed time horizon $T>0$, we model these movements via cylinders in $\mathbb {R}^d \times [0,T]$. In this work, we derive central limit theorems for functionals of the union of these cylinders. The volume and the number of isolated cylinders and the Euler characteristic of the random set are considered and give an answer to the achievable throughput, the availability of nodes, and the topological structure of the network.
In the classical framework, a random walk on a group is a Markov chain with independent and identically distributed increments. In some sense, random walks are time and space homogeneous. This paper is devoted to a class of inhomogeneous random walks on
$\mathbb{Z}^d$
termed ‘Markov additive processes’ (also known as Markov random walks, random walks with internal degrees of freedom, or semi-Markov processes). In this model, the increments of the walk are still independent but their distributions are dictated by a Markov chain, termed the internal Markov chain. While this model is largely studied in the literature, most of the results involve internal Markov chains whose operator is quasi-compact. This paper extends two results for more general internal operators: a local limit theorem and a sufficient criterion for their transience. These results are thereafter applied to a new family of models of drifted random walks on the lattice
$\mathbb{Z}^d$
.
The problem of optimally scaling the proposal distribution in a Markov chain Monte Carlo algorithm is critical to the quality of the generated samples. Much work has gone into obtaining such results for various Metropolis–Hastings (MH) algorithms. Recently, acceptance probabilities other than MH are being employed in problems with intractable target distributions. There are few resources available on tuning the Gaussian proposal distributions for this situation. We obtain optimal scaling results for a general class of acceptance functions, which includes Barker’s and lazy MH. In particular, optimal values for Barker’s algorithm are derived and found to be significantly different from that obtained for the MH algorithm. Our theoretical conclusions are supported by numerical simulations indicating that when the optimal proposal variance is unknown, tuning to the optimal acceptance probability remains an effective strategy.
We consider the random splitting and aggregating of Hawkes processes. We present the random splitting schemes using the direct approach for counting processes, as well as the immigration–birth branching representations of Hawkes processes. From the second scheme, it is shown that random split Hawkes processes are again Hawkes. We discuss functional central limit theorems (FCLTs) for the scaled split processes from the different schemes. On the other hand, aggregating multivariate Hawkes processes may not necessarily be Hawkes. We identify a necessary and sufficient condition for the aggregated process to be Hawkes. We prove an FCLT for a multivariate Hawkes process under a random splitting and then aggregating scheme (under certain conditions, transforming into a Hawkes process of a different dimension).
Given a graphon
$W$
and a finite simple graph
$H$
, with vertex set
$V(H)$
, denote by
$X_n(H, W)$
the number of copies of
$H$
in a
$W$
-random graph on
$n$
vertices. The asymptotic distribution of
$X_n(H, W)$
was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where
$H$
is a clique. In this paper, we extend this result to any fixed graph
$H$
. Towards this we introduce a notion of
$H$
-regularity of graphons and show that if the graphon
$W$
is not
$H$
-regular, then
$X_n(H, W)$
has Gaussian fluctuations with scaling
$n^{|V(H)|-\frac{1}{2}}$
. On the other hand, if
$W$
is
$H$
-regular, then the fluctuations are of order
$n^{|V(H)|-1}$
and the limiting distribution of
$X_n(H, W)$
can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from
$W$
. Our proofs use the asymptotic theory of generalised
$U$
-statistics developed by Janson and Nowicki [22]. We also investigate the structure of
$H$
-regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also
$H$
-regular graphons
$W$
for which both the Gaussian or the non-Gaussian components are degenerate, that is,
$X_n(H, W)$
has a degenerate limit even under the scaling
$n^{|V(H)|-1}$
. We give an example of this degeneracy with
$H=K_{1, 3}$
(the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.
Let
$\mu $
be a probability measure on
$\mathrm {GL}_d(\mathbb {R})$
, and denote by
$S_n:= g_n \cdots g_1$
the associated random matrix product, where
$g_j$
are i.i.d. with law
$\mu $
. Under the assumptions that
$\mu $
has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry–Esseen bound with the optimal rate
$O(1/\sqrt n)$
for the coefficients of
$S_n$
, settling a long-standing question considered since the fundamental work of Guivarc’h and Raugi. The local limit theorem for the coefficients is also obtained, complementing a recent partial result of Grama, Quint and Xiao.
We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.
We consider the problem of group testing (pooled testing), first introduced by Dorfman. For nonadaptive testing strategies, we refer to a nondefective item as “intruding” if it only appears in positive tests. Such items cause misclassification errors in the well-known COMP algorithm and can make other algorithms produce an error. It is therefore of interest to understand the distribution of the number of intruding items. We show that, under Bernoulli matrix designs, this distribution is well approximated in a variety of senses by a negative binomial distribution, allowing us to understand the performance of the two-stage conservative group testing algorithm of Aldridge.
Let X be a continuous-time strongly mixing or weakly dependent process and let T be a renewal process independent of X. We show general conditions under which the sampled process
$(X_{T_i},T_i-T_{i-1})^{\top}$
is strongly mixing or weakly dependent. Moreover, we explicitly compute the strong mixing or weak dependence coefficients of the renewal sampled process and show that exponential or power decay of the coefficients of X is preserved (at least asymptotically). Our results imply that essentially all central limit theorems available in the literature for strongly mixing or weakly dependent processes can be applied when renewal sampled observations of the process X are at our disposal.
We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of non-negative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We study the analytic properties of this Halász–Székely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of non-negative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converge to the Halász–Székely barycenter of the corresponding distribution.
We study large-deviation probabilities of Telecom processes appearing as limits in a critical regime of the infinite-source Poisson model elaborated by I. Kaj and M. Taqqu. We examine three different regimes of large deviations (LD) depending on the deviation level. A Telecom process
$(Y_t)_{t \ge 0}$
scales as
$t^{1/\gamma}$
, where t denotes time and
$\gamma\in(1,2)$
is the key parameter of Y. We must distinguish moderate LD
${\mathbb P}(Y_t\ge y_t)$
with
$t^{1/\gamma} \ll y_t \ll t$
, intermediate LD with
$ y_t \approx t$
, and ultralarge LD with
$ y_t \gg t$
. The results we obtain essentially depend on another parameter of Y, namely the resource distribution. We solve completely the cases of moderate and intermediate LD (the latter being the most technical one), whereas the ultralarge deviation asymptotics is found for the case of regularly varying distribution tails. In all the cases considered, the large-deviation level is essentially reached by the minimal necessary number of ‘service processes’.
The $q$-coloured Delannoy numbers $D_{n,k}(q)$ count the number of lattice paths from $(0,\,0)$ to $(n,\,k)$ using steps $(0,\,1)$, $(1,\,0)$ and $(1,\,1)$, among which the $(1,\,1)$ steps are coloured with $q$ colours. The focus of this paper is to study some analytical properties of the polynomial matrix $D(q)=[d_{n,k}(q)]_{n,k\geq 0}=[D_{n-k,k}(q)]_{n,k\geq 0}$, such as the strong $q$-log-concavity of polynomial sequences located in a ray or a transversal line of $D(q)$ and the $q$-total positivity of $D(q)$. We show that the zeros of all row sums $R_n(q)=\sum \nolimits _{k=0}^{n}d_{n,k}(q)$ are in $(-\infty,\, -1)$ and are dense in the corresponding semi-closed interval. We also prove that the zeros of all antidiagonal sums $A_n(q)=\sum \nolimits _{k=0}^{\lfloor n/2 \rfloor }d_{n-k,k}(q)$ are in the interval $(-\infty,\, -1]$ and are dense there.
We establish central limit theorems for an action of a group $G$ on a hyperbolic space $X$ with respect to the counting measure on a Cayley graph of $G$. Our techniques allow us to remove the usual assumptions of properness and smoothness of the space, or cocompactness of the action. We provide several applications which require our general framework, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds.
This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of the Stein equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called
$d_\pi$
, stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the point processes on arbitrary collections of disjoint sets. As applications, the multivariate Poisson approximation of the sum of m-dependent Bernoulli random vectors, the Poisson process approximation of point processes of U-statistic structure, and the Poisson process approximation of point processes with Papangelou intensity are considered. Our bounds in
$d_\pi$
are as good as those already available in the literature.
We introduce a general two-colour interacting urn model on a finite directed graph, where each urn at a node reinforces all the urns in its out-neighbours according to a fixed, non-negative, and balanced reinforcement matrix. We show that the fraction of balls of either colour converges almost surely to a deterministic limit if either the reinforcement is not of Pólya type or the graph is such that every vertex with non-zero in-degree can be reached from some vertex with zero in-degree. We also obtain joint central limit theorems with appropriate scalings. Furthermore, in the remaining case when there are no vertices with zero in-degree and the reinforcement is of Pólya type, we restrict our analysis to a regular graph and show that the fraction of balls of either colour converges almost surely to a finite random limit, which is the same across all the urns.
Motivated by recent studies of big samples, this work aims to construct a parametric model which is characterized by the following features: (i) a ‘local’ reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term almost sure convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness-of-fit result for the limit probabilities. This triple purpose is achieved by the introduction of a new variant of the Eggenberger–Pólya urn, which we call the rescaled Pólya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited by the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.