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2001 Pitman's $2M-X$ Theorem for Skip-Free Random Walks with Markovian Increments
B. Hambly, James Martin, Neil O'Connell
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Electron. Commun. Probab. 6: 73-77 (2001). DOI: 10.1214/ECP.v6-1036

Abstract

Let $(\xi_k, k\ge 0)$ be a Markov chain on ${-1,+1}$ with $\xi_0=1$ and transition probabilities $P(\xi_{k+1}=1| \xi_k=1)=a>b=P(\xi_{k+1}=-1| \xi_k=-1)$. Set $X_0=0$, $X_n=\xi_1+\cdots +\xi_n$ and $M_n=\max_{0\le k\le n}X_k$. We prove that the process $2M-X$ has the same law as that of $X$ conditioned to stay non-negative.

Citation

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B. Hambly. James Martin. Neil O'Connell. "Pitman's $2M-X$ Theorem for Skip-Free Random Walks with Markovian Increments." Electron. Commun. Probab. 6 73 - 77, 2001. https://doi.org/10.1214/ECP.v6-1036

Information

Accepted: 21 August 2001; Published: 2001
First available in Project Euclid: 19 April 2016

zbMATH: 0985.60070
MathSciNet: MR1855343
Digital Object Identifier: 10.1214/ECP.v6-1036

Subjects:
Primary: 60J10
Secondary: 60J27, 60J45, 60K25

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