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Continuity of random sequences and approximation of Markov chains

Published online by Cambridge University Press:  01 July 2016

V. V. Kalashnikov  and
S. A. Anichkin
V. V. Kalashnikov*
Affiliation:
Institute for Systems Studies
S. A. Anichkin*
Affiliation:
Moscow Physico-Technical Institute
*
∗Postal address: Institute for Systems Studies, 29 Ryleyev St., 119034 Moscow, U.S.S.R.
∗∗Postal address: Department of Applied Mathematics and Control, Moscow Physico-Technical Institute, 141700 g. Dolgoprudnyi, U.S.S.R.
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Abstract

We derive conditions for the time-uniform continuity of random sequences with respect to variations of governing parameters, and also obtain some estimates of the modulus of continuity. We then apply the results to find conditions of continuity for Markov chains with arbitrary state space, and to construct finite approximations to them.

Keywords

MARKOV CHAINSCONTINUITYMETRICESTIMATECOMPACTNESSWEAK CONVERGENCE
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 2 , June 1981 , pp. 402 - 414
Copyright
Copyright © Applied Probability Trust 1981 

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References

[1] Akhmarov, I. (1979) On the rate of convergence in ergodicity and continuity theorems for multichannel queueing systems. Theory Prob. Appl. 24, 423–429.CrossRefGoogle Scholar
[2] Borovkov, A. A. (1978) Ergodic theorems and stability for a class of stochastic equations and their applications. Theory Prob. Appl. 23, 227–247.Google Scholar
[3] Dudley, R. M. (1968) Distances of probability measures and random variables. Ann. Math. Statist. 39, 1563–1572.Google Scholar
[4] Dugué, D. (1958) Traité de statistique théorique et appliqué. Masson, Paris.Google Scholar
[5] Kalashnikov, V. V. (1973) The property of γ-reflexivity for Markov sequences. Soviet Math. Dokl. 14, 1869–1873.Google Scholar
[6] Kalashnikov, V. V. (1978) Qualitative Analysis of the Behaviour of Complex Systems by Trial Functions (in Russian). Nauka, Moscow.Google Scholar
[7] Kalashnikov, V. V. (1978) A decision of the approximation problem for a denumerable Markov chain (in Russian). Izv. Akad. Nauk. SSSR Tehn. Kibernet. N3, 92–95.Google Scholar
[8] Kalashnikov, V. V. (1979) Estimates of stability for renovative processes (in Russian). Izv. Akad. Nauk SSSR Tehn. Kibernet. N5, 85–89.Google Scholar
[9] Karr, A. F. (1975) Weak convergence of a sequence of Markov chains. Z. Wahrscheinlichkeitsth. 33, 41–48.Google Scholar
[10] Prokhorov, Yu. V. (1956) Convergence of random processes and limit theorems in probability theory. Theory Prob. Appl. 1, 157–214.CrossRefGoogle Scholar
[11] Tsitsiashvili, G. Sh. (1975) Piecewise linear Markov chains and analysis of their stability. Theory Prob. Appl. 20, 337–350.Google Scholar
[12] Zhilin, V. A. and Kalashnikov, V. V. (1979) Estimates of stability for regenerative processes and their applications to priority queues. Izv. Akad. Nauk. SSSR. Tehn. Kibernet. N4, 94–101.Google Scholar
[13] Zolotarev, V. M. (1975) On the continuity of stochastic sequences generated by recurrent processes. Theory Prob. Appl. 20, 819–832.Google Scholar
[14] Zolotarev, V. M. (1977) General problems of the stability of mathematical models. Proc. 41st Session Internat. Statist. Inst., New Delhi. Google Scholar