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 $(xi_k, kge 0)$ 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 $M_n=max_{0le
kle n}X_k$. We prove that the process $2M-X$ has the same law as that
of $X$ conditioned to stay non-negative.
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