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

   
@article{ECP1036,
	author = {B. Hambly and James Martin and Neil O'Connell},
	title = {Pitman's $2M-X$ Theorem for Skip-Free Random Walks with Markovian Increments},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {6},
	year = {2001},
	keywords = {Pitman's representation, three-dimensional Bessel process,telegrapher's equation, queue, Burke's theorem, quasireversibility.},
	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.},
	pages = {no. 7, 73-77},
	issn = {1083-589X},
	doi = {10.1214/ECP.v6-1036},    
        url = {http://ecp.ejpecp.org/article/view/1036}}