Growth of Random Walks Conditioned to Stay Positive

Abstract

For random walk $S_k = \sum^k_{i = 1} \xi_i$ let $T$ be the hitting time of the lower half plane. By conditioning the process $\{S_k\}^n_{k = 1}$ relative to $\lbrack T > n\rbrack$ we create a "random walk conditioned to stay positive." Sample paths of such processes tend to grow rather quickly. In studying this growth we find that except for a set of probability $\epsilon$ all such sample paths have as lower bounds any sequence of the form $\{\delta k^\eta\}^n_{k = 1}$ where $\eta \in (0, 1/2)$ and $\delta < \delta(\epsilon, \eta)$. Applications of this result to sample path behavior of a random walk as it approaches or leaves the lowest of its first $n$ values are also given.