Transition phenomena for ladder epochs of random walks with small negative drift

TL;DR

This paper investigates the asymptotic behavior of ladder epochs in a family of random walks with small negative drift, revealing how the probability of the walk remaining positive evolves as the drift approaches zero.

Contribution

It provides new asymptotic results for the probability and moments of ladder epochs in random walks with diminishing negative drift.

Findings

Asymptotic behavior of $ extbf{P}( au^{(a)}>n)$ as $a o 0$ and $n o \infty$

Growth rates of moments of ladder epochs $ au^{(a)}$

Characterization of transition phenomena in ladder epochs for small negative drift

Abstract

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 when $n=n(a)\to\infty$. As a consequence we obtain also the growth rates of the moments of $\tau^{(a)}$.