Summary
Consider a discrete time parameter Markov Process with stationary probability functions, a general state spaceX and the Harris recurrence condition. This then implies the existence and essential uniqueness of a sigma-finite stationary measureπ. It is also assumed that the class of measurable sets ∇ contains single point sets. LetP (m)(x, S) denote them-step transition probability fromx toS∃∇ andp (m)(x, ·), the component ofP (m)(x, ·) which is absolutely continuous with respect toπ. Let ℒ=C: C∃∇, for some\(\mathop {\inf }\limits_{x,y \in C} p^{(n)} (x,y) > 0\} \) and\( = \{ n:\mathop {\inf ,}\limits_{x,y \in C} p^{(n)} (x,y) > 0,C \in \)ℒ}. The paper here presented contains theorems of which the following is typical:Theorem: LetS∃ℐ withπ(S)>0, measurableB⊃S, π(B)>0 andq∃B with\(\mathop {\lim }\limits_{m \to \infty } \int\limits_B {f(x)P^{(m + 1)} (y,dx)/P^{(m)} (q,B)} = \int\limits_B {f(x)\pi } (dx)/\pi ({\rm B})\) uniformly iny, y∃B for all non-negative measurable f. Then for all measurableA⊃S withπ(A)>0,k=0,±1, ±2,...\(\mathop {\lim }\limits_{m \to \infty } P^{(m + k)} (x,A)/P^{(m)} (q,B) = \pi ({\rm A})/\pi ({\rm B})\) in measureπ onS. If the g.c.d. (ℐ)=1 and π′ ≪π withπ′(X)<∞ then the above limit holds in measureπ′ onX.
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This research was partially supported by National Science Foundation Grant Gp-3906.
The author would like to convey his appreciation to Professor Steven Orey for his invaluable guidance, advice and encouragement.
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Levitan, M.L. Some ratio limit Theorems for a general state space Markov Process. Z. Wahrscheinlichkeitstheorie verw Gebiete 15, 29–50 (1970). https://doi.org/10.1007/BF01041973
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DOI: https://doi.org/10.1007/BF01041973