ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS |
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Authors: | Masaaki Kijima |
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Affiliation: | Graduate School of Systems Management, The University of Tsukuba Bunkyo-ku, Tokyo 112, Japan |
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Abstract: | Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (qij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., qij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability pij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p0N(t) (p(m)(N0)(t)), 0 ≤ m ≤N, where f(m)(t) denotes the mth derivative of a function f(t) with f(0)(t) =f(t). If X(t) is a birth-death process, then pij(t) is represented as a linear combination of p0N(m)(t), 0 ≤m≤N - |i-j|. |
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Keywords: | Transition probability skip-free Markov chain birth-death process |
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