Renewal Theory for Functionals of a Markov Chain with General State Space

Abstract

We prove an analogue of Blackwell's renewal theorem or the "key renewal theorem" and the existence of the limit distribution of the residual waiting time in the following setup: X0,X1,⋯ is a Markov chain with separable metric state space and u0,u1,⋯ is a sequence of random variables, such that the conditional distribution of ui, given all Xj and ul,l≠i, depends on Xi and Xi+1 only. Here the Vn≡∑n−10ui,n≧1, take the role of the partial sums of independent identically distributed random variables in ordinary renewal theory. E.g. the key renewal theorem in this setup states that limt→∞E{∑∞n=0g(Xn,t−Vn)∣X0=x} exists for suitable g(∙,∙), and is independent of x.