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Recurrence for Markov processes on N lines

Published online by Cambridge University Press:  14 July 2016

Mark Pinsky
Mark Pinsky*
Affiliation:
Northwestern University, Evanston, Illinois
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Extract

Let Λ = R1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ(t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 4 , December 1971 , pp. 724 - 730
Copyright
Copyright © Applied Probability Trust 1971 

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References

[1] Dynkin, E. B. (1965) Markov Processes. Academic Press, New York.Google Scholar
[2] Fukushima, M. and Hitsuda, M. (1967) On a class of Markov processes taking values on lines and the central limit theorem. Nagoya Math. J. 30, 7–56.CrossRefGoogle Scholar
[3] Hersh, R. and Pinsky, M. (1971) Random evolutions are asymptotically Gaussian. Comm. Pure Appl. Math. (To appear).CrossRefGoogle Scholar
[4] Hitsuda, M. and Shimizu, A. (1970) The central limit theorem for additive functionals of Markov processes and the weak convergence to Wiener measure. J. Math. Soc. Japan 22, 551–566.Google Scholar
[5] Ito, K. (1959) Lectures on Stochastic Processes. Tata Institute Bombay.Google Scholar
[6] Pinsky, M. (1968) Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain. Z. Wahrscheinlichkeitsth. 9, 101–111.Google Scholar