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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’.
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.
We study a stochastic compartmental susceptible–infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.
It is well known that each statistic in the family of power divergence statistics, across n trials and r classifications with index parameter
$\lambda\in\mathbb{R}$
(the Pearson, likelihood ratio, and Freeman–Tukey statistics correspond to
$\lambda=1,0,-1/2$
, respectively), is asymptotically chi-square distributed as the sample size tends to infinity. We obtain explicit bounds on this distributional approximation, measured using smooth test functions, that hold for a given finite sample n, and all index parameters (
$\lambda>-1$
) for which such finite-sample bounds are meaningful. We obtain bounds that are of the optimal order
$n^{-1}$
. The dependence of our bounds on the index parameter
$\lambda$
and the cell classification probabilities is also optimal, and the dependence on the number of cells is also respectable. Our bounds generalise, complement, and improve on recent results from the literature.
Branching-stable processes have recently appeared as counterparts of stable subordinators, when addition of real variables is replaced by branching mechanisms for point processes. Here we are interested in their domains of attraction and describe explicit conditions for a branching random walk to converge after a proper magnification to a branching-stable process. This contrasts with deep results obtained during the past decade on the asymptotic behavior of branching random walks and which involve either shifting without rescaling, or demagnification.
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.
The rich-get-richer rule reinforces actions that have been frequently chosen in the past. What happens to the evolution of individuals’ inclinations to choose an action when agents interact? Interaction tends to homogenize, while each individual dynamics tends to reinforce its own position. Interacting stochastic systems of reinforced processes have recently been considered in many papers, in which the asymptotic behavior is proven to exhibit almost sure synchronization. In this paper we consider models where, even if interaction among agents is present, absence of synchronization may happen because of the choice of an individual nonlinear reinforcement. We show how these systems can naturally be considered as models for coordination games or technological or opinion dynamics.
We prove polynomial ergodicity for the one-dimensional Zig-Zag process on heavy-tailed targets and identify the exact order of polynomial convergence of the process when targeting Student distributions.
$U{\hbox{-}}\textrm{max}$
statistics were introduced by Lao and Mayer in 2008. Such statistics are natural in stochastic geometry. Examples are the maximal perimeters and areas of polygons and polyhedra formed by random points on a circle, ellipse, etc. The main method to study limit theorems for
$U{\hbox{-}}\textrm{max}$
statistics is via a Poisson approximation. In this paper we consider a general class of kernels defined on a circle, and we prove a universal limit theorem with the Weibull distribution as a limit. Its parameters depend on the degree of the kernel, the structure of its points of maximum, and the Hessians of the kernel at these points. Almost all limit theorems known so far may be obtained as simple special cases of our general theorem. We also consider several new examples. Moreover, we consider not only the uniform distribution of points but also almost arbitrary distribution on a circle satisfying mild additional conditions.
Consider a finite or infinite collection of urns, each with capacity r, and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain r balls. When
$r=1$
, this is the number of balls landing in non-empty urns, which has been studied in the past. Our aim here is to use martingale methods to study the asymptotics of the overflow in the general situation, i.e. for arbitrary r. In particular, we provide sufficient conditions for both Poissonian and normal asymptotics.
This article examines large-time behaviour of finite-state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) of the time required for convergence of the empirical measure process of the N-particle system to its invariant measure; we show that when time is of the order
$\exp\{N\Lambda\}$
for a suitable constant
$\Lambda > 0$
, the process has mixed well and it is close to its invariant measure. We then obtain large-N asymptotics of the second-largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as
$\exp\{{-}N\Lambda\}$
. The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large-time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain ‘entropy’ function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.