Stochastic Ordering for Bernoulli and Normal Random Walks

TL;DR

This paper establishes stochastic ordering results for the absolute values of Bernoulli and normal random walks, showing how these processes compare as their parameters vary, and applies these findings to stopping times.

Contribution

It provides new stochastic ordering results for Bernoulli and normal random walks, including their absolute values, and extends these results to Brownian motion.

Findings

Absolute value processes are stochastically ordered by parameter p in Bernoulli walks.

Normal random walks exhibit similar stochastic ordering with respect to mean μ.

Applications include deriving stochastic orderings for stopping times of these processes.

Abstract

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.