Related papers: Local limit theorem for the maximum of a random wa…

We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $\mathbb{P}[X \geq…

Consider the random walk $S_n=\xi_1+...+\xi_n$ with independent and identically distributed increments and negative mean $\mathbf E\xi=-m<0$. Let $M=\sup_{0\le i} S_i$ be the supremum of the random walk. In this note we present derivation…

We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to…

We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum…

Let $(\xi_k)$ and $(\eta_k)$ be infinite independent samples from different distributions. We prove a functional limit theorem for the maximum of a perturbed random walk $\underset{0\leq k\leq n}{\max}\,(\xi_1+\ldots+\xi_k+\eta_{k+1})$ in a…

We give an elementary probabilistic proof of Veraverbeke's Theorem for the asymptotic distribution of the maximum of a random walk with negative drift and heavy-tailed increments. The proof gives insight into the principle that the maximum…

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time $n$. Assuming that the moment of order $2+\delta$ is…

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…

We study downward deviations of the maximum local time of the discrete-time simple random walk on $\mathbb{Z}^d$, $d\ge 3$. In our previous paper \cite{li2026ldmaxlocal}, the corresponding upper bound was established, while the matching…

We consider random walks, say $W_n=(M_0, M_1,\dots, M_n)$, of length $n$ starting at 0 and based on the martingale sequence $M_k$ with differences $X_m=M_m-M_{m-1}$. Assuming that the differences are bounded, $|X_m|\leq 1$, we solve the…

This paper concerns a scaling limit of a one-dimensional random walk $S^x_n$ started from $x$ on the integer lattice conditioned to avoid a non-empty finite set $A$, the random walk being assumed to be irreducible and have zero mean.…

In this paper we study the asymptotic behavior of the (skew) Macdonald and Jack symmetric polynomials as the number of variables grows to infinity. We characterize their limits in terms of certain variational problems. As an intermediate…

Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $\alpha\in[0,\infty)$ and let $L_n(\alpha)$ be the spatial sum of the $\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range,…

We consider a one-dimensional random walk $S_n$ with i.i.d. increments with zero mean and finite variance. We study the asymptotic expansion for the tail distribution $\mathbf P(\tau_x>n)$ of the first passage times…

A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being $+1$ or -1, equally likely. The other families cited in the title are Bernoulli random walks under various conditionings. A peak…

We consider a recurrent random walk on a rooted tree in random environment given by a branching random walk. Up to the first return to the root, its edge local times form a Multi-type Galton-Watson tree with countably infinitely many types.…

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that…

Let $F$ be a distribution function on the integer lattice $\mathbb{Z}$ and $S=(S_n)$ the random walk with step distribution $F$. Suppose $S$ is oscillatory and denote by $U_{\rm a}(x)$ and $u_{\rm a}(x)$ the renewal function and sequence,…

Given a finite-range random walk on a finitely generated free group , what is the asymptotic behaviour, as the number of steps goes to infinity, of the sequence of probabilities that the random walk is at a given element of the group? In…

We derive subexponential tail asymptotics for the distribution of the maximum of a compound renewal process with linear component and of a L\'evy process, both with negative drift, over random time horizon $\tau$ that does not depend on the…