Stochastic Ordering for Bernoulli and Normal Random Walks

Let $(S_n^p)_{n\geq 0}$ be a Bernoulli random walk where each of the independent increments is either $1$ or $-1$ with probabilities $p$ and $1-p$. For $p'$ and $p'' \in [0,1]$ with $|p'-1/2|>|p''-1/2|$, we show that $(|S_n^{p''}|)_{n\geq 0}$ is stochastically smaller than $(|S_n^{p'}|)_{n\geq 0}$. In other words, $(|S_n^{p}|)_{n\geq 0}$ 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.