Related papers: Explicit stationary distribution of the $(L,1)$-re…
We study the distribution of the maximum $M$ of a random walk whose increments have a distribution with negative mean and belonging, for some $\gamma>0$, to a subclass of the class $\mathcal{S}_\gamma$--see, for example, Chover, Ney, and…
A semi-martingale reflecting Brownian motion is a popular process for diffusion approximations of queueing models including their networks. In this paper, we are concerned with the case that it lives on the nonnegative half-line, but the…
Consider a reflected jump-diffusion on the positive half-line. Assume it is stochastically ordered. We apply the theory of Lyapunov functions and find explicit estimates for the rate of exponential convergence to the stationary…
We investigate a stationary random coefficient autoregressive process. Using renewal type arguments tailor-made for such processes, we show that the stationary distribution has a power-law tail. When the model is normal, we show that the…
This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such type of random walks are characterized by the fact that the one-step transition probabilities are functions of the…
In this article, we first give a comprehensive description of random walk (RW) problem focusing on self-similarity, dynamic scaling and its connection to diffusion phenomena. One of the main goals of our work is to check how robust the RW…
We consider a multi-type branching random walk with displacements that have either regularly varying or semi-exponential tails. We investigate the asymptotic behavior of the rightmost particle in irreducible and reducible regimes and…
Brownian motion in R 2 + with covariance matrix $\Sigma$ and drift $\mu$ in the interior and reflection matrix R from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in R 2 + is found…
We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, $f(\eta)$, characterized by a L\'evy index $\mu \in (0,2]$, which includes standard random walks ($\mu=2$) and L\'evy flights ($0<\mu<2$). We…
In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, $G_n$, and the number of time steps, $L_n$, between the two highest positions of a Markovian one-dimensional random walker,…
We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the…
This paper is devoted to the asymptotic analysis of the reinforced elephant random walk (RERW) using a martingale approach. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and…
We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. It is assumed that this reflecting random walk has skip free transitions. We are concerned with its time reversed process assuming that the stationary…
Let $(G,\rho)$ be a stationary random graph, and use $B^G_{\rho}(r)$ to denote the ball of radius $r$ about $\rho$ in $G$. Suppose that $(G,\rho)$ has annealed polynomial growth, in the sense that $\mathbb{E}[|B^G_{\rho}(r)|] \leq O(r^k)$…
Distribution of loops in a one-dimensional random walk (RW), or, equivalently, neutral segments in a sequence of positive and negative charges is important for understanding the low energy states of randomly charged polymers. We investigate…
We study the distribution of occupation times for a one-dimensional random walk restricted to a finite interval by reflecting boundary conditions. At short times the classical bimodal distribution due to L\'evy is reproduced with walkers…
We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently…
We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as…
This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such type of random walks in the quarter plane are characterized by the fact that the one-step transition probabilities…
We review recent studies demonstrating a nonuniversal (continuously variable) survival exponent for history-dependent random walks, and analyze a new example, the hard movable partial reflector. These processes serve as a simplified models…