Related papers: Limit theorems for functionals of long memory line…

We consider a pure death process $(Z(t), t\ge0)$ with death rates $\lambda_n$ satisfying the condition $\sum_{n=2}^\infty \lambda_n^{-1}<\infty$ of coming from infinity, $Z(0)=\infty$, down to an absorbing state $n=1$. We establish limit…

Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in{\mathbb Z}^d)$ are two independent sequences of i.i.d. random variables with values in ${\mathbb Z}^d$ and…

We study the extent to which wide neural networks may be approximated by Gaussian processes when initialized with random weights. It is a well-established fact that as the width of a network goes to infinity, its law converges to that of a…

We consider a L\'evy process that starts from $x<0$ and conditioned on having a positive maximum. When Cram\'er's condition holds, we provide two weak limit theorems as $x\to -\infty$ for the law of the (two-sided) path shifted at the first…

Central limit theorems are established for the sum, over a spatial region, of observations from a linear process on a $d$-dimensional lattice. This region need not be rectangular, but can be irregularly-shaped. Separate results are…

In this paper we study the asymptotic behaviour of weighted random sums when the sum process converges stably in law to a Brownian motion and the weight process has continuous trajectories, more regular than that of a Brownian motion. We…

In this paper we prove a strong law of large numbers and its L^1-convergence counterpart for the process counted with a random characteristic in the context of self-similar fragmentation processes. This result extends a somewhat analogical…

We derive theorems which outline explicit mechanisms by which anomalous scaling for the probability density function of the sum of many correlated random variables asymptotically prevails. The results characterize general anomalous scaling…

In this paper we establish Functional Limit Theorems for the range of random walks in $\mathbb{Z}^d$ that are in the domain of attraction of a non-degenerate $\beta$-stable process in the weakly transient and recurrent regimes. These…

Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0, 1)$. The study of the growth rate of the product of consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with the…

In this article, we consider a sequence $(N_n)_{n \geq 1}$ of point processes, whose points lie in a subset $E$ of $\bR \verb2\2 \{0\}$, and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient…

We investigate branching processes in nearly degenerate varying environment, where the offspring distribution converges to the degenerate distribution at 1. Such processes die out almost surely, therefore, we condition on non-extinction or…

In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every aperiodic dynamical system, for every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and…

Consider a sequence X_k=\sum_{j=0}^{\infty}c_j\xi_{k-j}, k\geq 1, where c_j, j\geq 0, is a sequence of constants and \xi_j, -\infty <j<\infty, is a sequence of independent identically distributed (i.i.d.) random variables (r.v.s) belonging…

We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N= S_N(\alpha,x)$ be the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial point,…

Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience, social networks,…

We obtain the law of large numbers (LLN) and the central limit theorem (CLT) for weakly dependent non-stationary arrays of random fields with asymptotically unbounded moments. The weak dependence condition for arrays of random fields is…

Recently, for the joint partial sum and partial maxima processes constructed from linear processes with independent identically distributed innovations that are regularly varying with tail index $\alpha \in (0, 2)$, a functional limit…

In the paper we consider the partial sum process $\sum_{k=1}^{[nt]}X_k^{(n)}$, where $\{X_k^{(n)}, \ k\in Z\},\ n\ge 1,$ is a series of linear processes with innovations having heavy-tailed tapered distributions with tapering parameter…

The paper proves the Strong Law of Large Numbers for integral functionals of random fields with unboundedly increasing covariances. The case of functional data and increasing domain asymptotics is studied. Conditions to guarantee that the…