Stochastic Ordering for Bernoulli and Normal Random Walks

Abstract

Let $(S_n^p{)}_{n\ge 0}$ be a Bernoulli random walk where each of the independent increments is either $1$ or $-1$ with probabilities $p$ and $1-p$. For $p^{\prime }$ and $p^{\prime \prime }\in [0,1]$ with $|p^{\prime }-1/2|>|p^{\prime \prime }-1/2|$, we show that $(|S_n^{p^{\prime \prime }}|{)}_{n\ge 0}$ is stochastically smaller than $(|S_n^{p^{\prime }}|{)}_{n\ge 0}$. In other words, $(|S_n^p|{)}_{n\ge 0}$ is stochastically decreasing in $p\in [0,1/2]$ and increasing in $p\in [1/2,1]$. An analogous result is also given for the family of normal random walks indexed by $\mu \in R$ where each of the independent increments is normally distributed with common mean $\mu$ and variance $1$. Extension to Brownian motion then follows by a limiting argument. As an application, these results are used to easily derive stochastic ordering properties for stopping times of Bernoulli and normal random walks.