A theorem of Pitman type for simple random walks on \(Z^ d\) (Q913376)
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scientific article; zbMATH DE number 4147243
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A theorem of Pitman type for simple random walks on \(Z^ d\) |
scientific article; zbMATH DE number 4147243 |
Statements
A theorem of Pitman type for simple random walks on \(Z^ d\) (English)
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1989
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Let \(S_ n\) be a simple random walk on \({\mathbb{Z}}^ d\) starting at 0, \(S_ n^{(i)}\) its i-th component, \(M_ n^{(i)}\) the minimum of \(S_ k^{(i)}\), \(0\leq k\leq n\). Then \[ S_ n-2M_ n=(S_ n^{(1)}- 2M_ n^{(1)},...,S_ n^{(d)}-2M_ n^{(d)\quad}) \] is a Markov chain on \({\mathbb{Z}}^ d_+\) \((=set\) of points with non-negative integral coordinates).
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Pitman's theorem
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random walk
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