Related papers: Functional limit theorems for random walks

Open Quantum Random Walks, as developed in \cite{APSS}, are a quantum generalization of Markov chains on finite graphs or on lattices. These random walks are typically quantum in their behavior, step by step, but they seem to show up a…

We consider two dimensional random walks conditioned to stay in the positive quadrant. Assuming that the increments of the walk have finite second moments and that the drift vector is co-oriented with one of two axes, we construct positive…

A notion of convergence of excursion measures is introduced. It is proved that convergence of excursion measures implies convergence in law of the processes pieced together from excursions. This result is applied to obtain homogenization…

This paper is concerned with the limit theory of the extreme order statistics derived from random walks. We establish the joint convergence of the order statistics near the minimum of a random walk in terms of the Feller chains. Detailed…

Motivated by applications to the study of depth functions for tree-indexed random variables generated by point processes, we describe functional limit theorems for the intensity measure of point processes. Specifically, we establish uniform…

We first present a comprehensive review of various random walk metrics used in the literature and express them in a consistent framework. We then introduce fundamental tensor -- a generalization of the well-known fundamental matrix -- and…

We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension $d\ge 3$, started off either from a Poisson field or in equilibrium, when suitably normalized, converges to a…

We construct the conditional version of $k$ independent and identically distributed random walks on $\R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random…

We prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time. After deducing the limiting distribution for individual clusters of extremes, we present a new type of…

We consider random walks in dynamic random environments which arise naturally as spatial embeddings of ancestral lineages in spatial locally regulated population models. In particular, as the main result, we prove the quenched central limit…

We study limit distributions for random variables defined in terms of coefficients of a power series which is determined by a certain linear functional equation. Our technique combines the method of moments with the kernel method of…

Let $S=(S_k)_{k\geq 0}$ be a random walk on $\mathbb{Z}$ and $\xi=(\xi_{i})_{i\in\mathbb{Z}}$ a stationary random sequence of centered random variables, independent of $S$. We consider a random walk in random scenery that is the sequence of…

The original Donsker theorem says that a standard random walk converges in distribution to a Brownian motion in the space of continuous functions. It has recently been extended to enriched random walks and enriched Brownian motion. We use…

We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only positive dependence could be…

Two-dimensional (random) walks in cones are very natural both in combinatorics and probability theory: they are interesting for themselves and also because they are strongly related to other discrete structures. While walks restricted to…

In this note - starting from $d$-dimensional (with $d>1$) fuzzy vectors - we prove Donsker's classical invariance principle. We consider a fuzzy random walk ${S^*_n}=X^*_1+\cdots+X^*_n,$ where $\{X^*_i\}_1^{\infty}$ is a sequence of…

This paper proves a weak limit theorem for a one-dimensional split-step quantum walk and investigates the limit density function. In the density function, the difference between two Konno's functions appears.

In this paper, we prove central limit theorems for bias reduced estimators of the structure function of several multifractal processes, namely mutiplicative cascades, multifractal random measures, multifractal random walk and multifractal…

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of the random medium on the walk is small in…

We study the asymptotic behavior of zero-drift random walks confined to multidimensional convex cones, when the endpoint is close to the boundary. We derive a local limit theorem in the fluctuation regime.