Related papers: Angular asymptotics for multi-dimensional non-homo…

In this paper we generalize the result of directional transience from [SabotTournier10]. This enables us, by means of [Simenhaus07], [ZernerMerkl01] and [Bouchet12] to conclude that, on Z^d (for any dimension d), random walks in i.i.d.…

We investigate the tail distribution of the first exit time of Brownian motion with drift from a cone and find its exact asymptotics for a large class of cones. Our results show in particular that its exponential decreasing rate is a…

Consider a nearest-neighbor random walk with certain asymptotically zero drift on the positive half line. Let $M$ be the maximum of an excursion starting from $1$ and ending at $0.$ We study the distribution of $M$ and characterize its…

We consider the process $\{x-N(t):t\geq 0\}$, where $x\in\mathbb{R}_+$ and $\{N(t):t\geq 0\}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(\tau(x),A(x))$ where $\tau(x)$…

We consider multidimensional random walks in pyramids, which by definition are cones formed by finite intersections of half-spaces. The main object of interest is the survival probability $\mathbb{P}(\tau>n)$, $\tau$ denoting the first exit…

We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that…

We prove that every random walk in a uniformly elliptic random environment satisfying the cone mixing condition and a non-effective polynomial ballisticity condition with high enough degree has an asymptotic direction.

We study the exact asymptotics for the distribution of the first time $\tau_x$ a L\'evy process $X_t$ crosses a negative level $-x$. We prove that $\mathbf P(\tau_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$ as $t\to\infty$ for a certain function…

We investigate Martin boundary for a non-centered random walk on ${\mathbb Z}^d$ killed up on the time $\tau_\vartheta$ of the first exit from a convex cone with a vertex at $0$. The approach combines large deviation estimates, the ratio…

Given a Gaussian random walk (or a Wiener process), possibly with drift, observed through noise, we consider the problem of estimating its first-passage time $\tau_\ell$ of a given level $\ell$ with a stopping time $\eta$ defined over the…

We study a discrete-time random walk on the non-negative integers, such that when 0 is reached a jump occurs to an arbitrary location, with given probabilities. We obtain an asymptotic formula for the expected position at large times, in…

We study the asymptotic behaviour of random walks in i.i.d. random environments on $\Z^d$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when…

We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is…

We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0…

We prove sharp asymptotic estimates for the rate of escape of the two-dimensional simple random walk conditioned to avoid a fixed finite set. We derive it from asymptotics available for the continuous analogue of this process (cf…

We evaluate the limit distribution of the maximal excursion of a random walk in any dimension for homogeneous environments and for self-similar supports under the assumption of spherical symmetry. This distribution is obtained in closed…

For the perimeter length and the area of the convex hull of the first $n$ steps of a planar random walk, we study $n \to \infty$ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random…

Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length $n$ of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time $n$.…

Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability \pi. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that…

For a family of random walks $\{S^{(a)}\}$ satisfying $\mathbf{E}S_1^{(a)}=-a<0$ we consider ladder epochs $\tau^{(a)}=\min\{k\geq1: S_k^{(a)}<0\}$. We study the asymptotic, as $a\to0$, behaviour of $\mathbf{P}(\tau^{(a)}>n)$ in the case…