We consider a supercritical branching process (Zn, n ≥ 0) with offspring distribution (pk, k ≥ 0) satisfying p0 = 0 and p1 > 0. By applying the self-normalized large deviation of Shao (1997) for independent and identically distributed random variables, we obtain the self-normalized large deviation for supercritical branching processes, which is the self-normalized version of the result obtained by Athreya (1994). The self-normalized large deviation can also be generalized to supercritical multitype branching processes.

Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

In this paper we study the Robbins–Monro procedure Xn+1 = Xn - an-1Yn with some fixed number a > 0 and establish the moderate deviation principle of the process {Xn}.

In this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an α-stable law with 1 < α < 2. We prove that the derivative martingale Dn converges to a nontrivial limit D∞ under some regular conditions. We also study the additive martingale Wn and prove that n1/αWn converges in probability to a constant multiple of D∞.

By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and light-tailed increments. We determine the asymptotics for local probabilities for the area and prove a local central limit theorem for the duration of the excursion conditioned on the large values of its area.

We study the asymptotic tail behavior of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -½, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

Let $\unicode[STIX]{x1D707}$ be the projection on $[0,1]$ of a Gibbs measure on $\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$ (or more generally a Gibbs capacity) associated with a Hölder potential. The thermodynamic and multifractal properties of $\unicode[STIX]{x1D707}$ are well known to be linked via the multifractal formalism. We study the impact of a random sampling procedure on this structure. More precisely, let $\{{I_{w}\}}_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the finite dyadic words. Fix $\unicode[STIX]{x1D702}\in (0,1)$, and a sequence $(p_{w})_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ of independent Bernoulli variables of parameters $2^{-|w|(1-\unicode[STIX]{x1D702})}$. We consider the (very sparse) remaining values $\widetilde{\unicode[STIX]{x1D707}}=\{\unicode[STIX]{x1D707}(I_{w}):w\in \unicode[STIX]{x1D6F4}^{\ast },p_{w}=1\}$. We study the geometric and statistical information associated with $\widetilde{\unicode[STIX]{x1D707}}$, and the relation between $\widetilde{\unicode[STIX]{x1D707}}$ and $\unicode[STIX]{x1D707}$. To do so, we construct a random capacity $\mathsf{M}_{\unicode[STIX]{x1D707}}$ from $\widetilde{\unicode[STIX]{x1D707}}$. This new object fulfills the multifractal formalism, and its free energy is closely related to that of $\unicode[STIX]{x1D707}$. Moreover, the free energy of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ generically exhibits one first order and one second order phase transition, while that of $\unicode[STIX]{x1D707}$ is analytic. The geometry of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ is deeply related to the combination of approximation by dyadic numbers with geometric properties of Gibbs measures. The possibility to reconstruct $\unicode[STIX]{x1D707}$ from $\widetilde{\unicode[STIX]{x1D707}}$ by using the almost multiplicativity of $\unicode[STIX]{x1D707}$ and concatenation of words is discussed as well.

We give a simple proof of the strong law of large numbers with rates, assuming only finite variance. This note also serves as an elementary introduction to the theory of large deviations, assuming only finite variance, even when the random variables are not necessarily independent.

Consider a graph on randomly scattered points in an arbitrary space, with any two points x, y connected with probability ϕ(x, y). Suppose the number of points is large but the mean number of isolated points is O(1). We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.

Let ξ = (ξ1, . . ., ξm) be a negatively associated mean-zero random vector with components that obey the bound |ξi| ≤ B, i = 1, . . ., m, and whose sum W = ∑i=1mξi has variance 1. The bound d1(ℒ(W), ℒ(Z)) ≤ 5B - 5.2∑i≠ jσij is obtained, where Z has the standard normal distribution and d1(∙, ∙) is the L1 metric. The result is extended to the multidimensional case with the L1 metric replaced by a smooth functions metric. Applications to second-order stationary random fields with exponential decreasing covariance are also presented.

We discuss weak convergence of the number of busy servers in a G/G/∞ queue in the J1-topology on the Skorokhod space. We prove two functional limit theorems with random and nonrandom centering, thereby solving two open problems stated in Mikosch and Resnick (2006). A new integral representation for the limit Gaussian process is given.

We show that under some slight assumptions, the positive sojourn time of a product of symmetric processes converges towards ½ as the number of processes increases. Monotony properties are then exhibited in the case of symmetric stable processes, and used, via a recurrence relation, to obtain upper and lower bounds on the moments of the occupation time (in the first and third quadrants) for two-dimensional Brownian motion. Explicit values are also given for the second and third moments in the n-dimensional Brownian case.

We introduce a large and flexible class of discrete tempered stable distributions, and analyze the domains of attraction for both this class and the related class of positive tempered stable distributions. Our results suggest that these are natural models for sums of independent and identically distributed random variables with tempered heavy tails, i.e. tails that appear to be heavy up to a point, but ultimately decay faster.

A classical Pólya urn scheme is a Markov process where the evolution is encoded by a replacement matrix (Ri, j)1 ≤ i, j ≤ d. At every discrete time-step, we draw a ball uniformly at random, denote its colour c, and replace it in the urn together with Rc, j balls of colour j (for all 1 ≤ j ≤ d). We study multiple drawing Pólya urns, where the replacement rule depends on the random drawing of a set of m balls from the urn (with or without replacement). Many particular examples of this situation have been studied in the literature, but the only general results are due to Kuba and Mahmoud (2017). These authors proved second-order asymptotic results in the two-colour case, under the so-called balance and affinity assumptions, the latter being somewhat artificial. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to prove analogous results to Kuba and Mahmoud, but without the artificial affinity hypothesis, and, for the first time in the literature, in the d-colour case (d ≥ 3). We also provide some partial results in the two-colour nonbalanced case, the novelty here being that the only results for this case currently in the literature are for particular examples.

In 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_{1}+a_{2}=a_{3}+a_{4}$ with $a_{i}\in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.

In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.

We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.