ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01041973
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