Pitman's $2M-X$ Theorem for Skip-Free Random Walks with Markovian Increments

B. M. Hambly (University of Oxford)
James B. Martin (Cambridge University)
Neil O'Connell (BRIMS, HP Labs)

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.

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Pages: 73-77

Publication Date: August 21, 2001

DOI: 10.1214/ECP.v6-1036

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