Stochastic Ordering for Bernoulli and Normal Random Walks
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912716162072576 |
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| author | Hsiau, Shoou-Ren Yao, Yi-Ching |
| author_facet | Hsiau, Shoou-Ren Yao, Yi-Ching |
| contents | 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 $μ\in R$ where each of the independent increments is normally distributed with common mean $μ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic Ordering for Bernoulli and Normal Random Walks Hsiau, Shoou-Ren Yao, Yi-Ching Probability 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 $μ\in R$ where each of the independent increments is normally distributed with common mean $μ$ 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. |
| title | Stochastic Ordering for Bernoulli and Normal Random Walks |
| topic | Probability |
| url | https://arxiv.org/abs/2511.14200 |