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Equilibrium processes

Sidney C. Port

1966
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Transactions of the American Mathematical Society
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1. Introduction. Let (£,$) be a set with a c-field of subsets \J containing all one point sets, and let P(x, B) be a transition function of a Markov process with states in £. Assume that F has at least one cr-finite invariant measure X which we take as fixed throughout the discussion. In §2 we describe precisely how to construct a system of denumerably many independent Markov processes all having the same transition law P and whose initial positions are given by the Poisson process on £ with
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... n X(B) on B. There we establish the fundamental fact that this system maintains itself in macroscopic equilibrium; thus we call such a process an equilibrium process. This property was first established for systems of this type by Doob for Brownian motion and by Derman for countable state space Markov chains. Our purpose in this paper will be to investigate (1) the number of processes, Mn(B; r), whose occupation time in B is exactly r by time n; (2) the number of distinct processes, Ln(B), which are in B at least once by time n; (3) the number of processes, Jn(B), which are in B for a last time at time n ; and (4) the number of processes, A"(B), which are in B at time n, where B is always a transient set of finite, positive X measure. Previously these quantities were investigated for this system in the countable state space case by the author in [6] , and the results we obtain here will be extensions of those in [6] to the more general setting of this paper(x). In summary, then, we do the following. In §2 we describe the construction of the basic system(2), and in §3 we give some preliminary material on Markov processes having nontrivial dissipative part which is needed in the sequel. In §4 we show that Mn(B; r)/n converges with probability one to a constant Cr(B) and that if Cr(B)>0, then [M"(B,r) -EMn(B,r)](nCr(B))~1'2 is asymptotically normally distributed. As a corollary we show that L"(B)/n converges with probability one to a constant C(B) > 0, and that [Ln(B)-ELn(B)](nC(B)y1'2 Presented to the Society, January 25, 1566 under the title A system of denumerably many transient Markov chains; received by the editors February 8, 1966. (') In the countable case the results were established by probabilistic arguments using the A-dual process. For the processes considered here, there is, in general, no dual process, and a more analytic approach is needed. (2) While in the countable case the construction of an equilibrium process is evident, this is no longer true in the general state space case. Some care is required to guarantee that the desired quantities aremeasurable.

doi:10.1090/s0002-9947-1966-0198546-1
fatcat:htngbrfi2ndrjkl7ujljkiivge