Limiting distributions of functionals of Markov chains |
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| Authors: | Rajeeva L Karandikar Vidyadhar G Kulkarni |
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| Institution: | Center for Stochastic Processes, University of North Carolina, Chapel Hill, NC 27514, USA;Curriculum in Operations Research and Systems Analysis, University of North Carolina, Chapel Hill, NC 27514, USA |
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| Abstract: | Let \s{Xn, n ? 0\s} and \s{Yn, n ? 0\s} be two stochastic processes such that Yn depends on Xn in a stationary manner, i.e. P(Yn ? A\vbXn) does not depend on n. Sufficient conditions are derived for Yn to have a limiting distribution. If Xn is a Markov chain with stationary transition probabilities and Yn = f(Xn,..., Xn+k) then Yn depends on Xn is a stationary way. Two situations are considered: (i) \s{Xn, n ? 0\s} has a limiting distribution (ii) \s{Xn, n ? 0\s} does not have a limiting distribution and exits every finite set with probability 1. Several examples are considered including that of a non-homogeneous Poisson process with periodic rate function where we obtain the limiting distribution of the interevent times. |
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| Keywords: | Markov chains limiting distributions periodic nonhomogeneous Poisson processes |
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