Related papers: Heavy traffic and heavy tails for the maximum of a…

For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka < 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ for asymptotically stable…

Consider a family of $\Delta$-latticed aperiodic random walks $\{S^{(a)},0\le a\le a_0\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose that $\sup_{a\le a_0}\mathbf{E}[(X^{(a)})^2]<\infty$ and $\sup_{a\le…

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…

Consider a random walk $S=(S_n:n\geq 0)$ that is ``perturbed'' by a stationary sequence $(\xi_n:n\geq 0)$ to produce the process $(S_n+\xi_n:n\geq0)$. This paper is concerned with computing the distribution of the all-time maximum…

This paper studies the asymptotic behavior of the steady-state waiting time, W_infty, of the M/G/1 queue with subexponenential processing times for different combinations of traffic intensities and overflow levels. In particular, we provide…

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…

This paper provides a detailed description for the asymptotics of exponential functionals of random walks with light/heavy tails. We give the convergence rate based on the key observation that the asymptotics depends on the sample paths…

We study conditions under which $P(S_\tau>x)\sim P(M_\tau>x)\sim E\tau P(\xi_1>x)$ as $x\to\infty$, where $S_\tau$ is a sum $\xi_1+...+\xi_\tau$ of random size $\tau$ and $M_\tau$ is a maximum of partial sums $M_\tau=\max_{n\le\tau}S_n$.…

Two of the most popular approximations for the distribution of the steady-state waiting time, $W_{\infty}$, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic…

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 sums $S_n=\xi_1+\cdots+\xi_n$ of independent identically distributed random variables. We do not assume that the $\xi$'s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the…

Let $\xi_1,\xi_2,\ldots$ be independent, identically distributed random variables with infinite mean $\mathbf E[|\xi_1|]=\infty.$ Consider a random walk $S_n=\xi_1+\cdots+\xi_n$, a stopping time $\tau=\min\{n\ge 1: S_n\le 0\}$ and let…

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…

We study the asymptotics for the maximum on a random time interval of a random walk with a long-tailed distribution of its increments and negative drift. We extend to a general stopping time a result by Asmussen (1998), simplify its proof,…

Let $\xi_1, \xi_2,\ldots$ be a sequence of independent and identically distributed random variables with zero mean, finite second moment and regularly varying right distribution tail. Motivated by a stop-loss insurance model, we consider a…

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…

Consider $M_n$ the maximal position at generation $n$ of a supercritical branching random walk. A\"id\'ekon (2013) obtained and described the convergence in law, as time $n$ goes to infinity, of $M_n-m_n$, where $m_n$ is an explicit…

The busy period for a queue is cast as the area swept under the random walk until it first returns to zero, $B$. Encompassing non-i.i.d. increments, the large-deviations asymptotics of $B$ is addressed, under the assumption that the…

We study the asymptotic probability that a random walk with heavy-tailed increments crosses a high boundary on a random time interval. We use new techniques to extend results of Asmussen [Ann. Appl. Probab. 8 (1998) 354-374] to completely…

Let $S_n =X_1+\cdots +X_n$ be an irreducible random walk (r.w.) on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments $X_n$. We obtain an upper and lower bounds of the potential function, $a(x)$, of…