LIMIT THEOREMS FOR ADDITIVE FUNCTIONALSOF A MARKOV CHAIN

摘要

Consider a Markov chain {X_n }_n>0with an ergodic probability mea-sure π. Let ψ be a function on the state space of the chain, with a-tailswith respect to π, α∈(0, 2). We find sufficient conditions on the probabil-ity transition to prove convergence in law of N~(1/α)Σ_n~Nψ(X_n)to an a-stablelaw. A "martingale approximation" approach and a "coupling" approach givetwo different sets of conditions. We extend these results to continuous timeMarkov jump processes X_t,whose skeleton chain satisfies our assumptions.If waiting times between jumps have finite expectation, we prove convergenceof N-11' ∫_0~(Nt)V (X_s) dsto a stable process. The result is applied to show thatan appropriately scaled limit of solutions of a linear Boltzman equation is asolution of the fractional diffusion equation.