Pitman's 2M - X theorem for skip-free random walks with Markovian increments

Journal article

Let (ξ k, k ≥ 0) be a Markov chain on {-1, +1} with ξ 0 = 1 and transition probabilities P(ξ k+1 = 1|ξ k = 1) = a and P(ξ k+1 = -1|ξ k = -1) = b < a. Set X 0 = 0, X n = ξ 1+⋯+ξ n and M n = max 0≤k≤n X k. We prove that the process 2M - X has the same law as that of X conditioned to stay non-negative.

Authors: Hambly, B; Martin, J; O'Connell, N

• Journal: Electronic Communications in Probability
• Volume: 6
• Pages: 73-77
• Publication date: 2001-08-21
• ISSN: 1083-589X
• Language: English
• Keywords: Burke's theorem; Pitman's representation; Telegrapher's equation; Three-dimensional Bessel process; Quasireversibility; Queue