On the functional central limit theorem and the law of the iterated logarithm for Markov processes

Abstract

Let Xt:t≧0 be an ergodic stationary Markov process on a state space S. If  is its infinitesimal generator on L2(S, dm), where m is the invariant probability measure, then it is shown that for all f in the range of {Mathematical expression} converges in distribution to the Wiener measure with zero drift and variance parameter σ2 =-2〈f, g〉=-2〈Âg, g〉 where g is some element in the domain of  such that Âg=f (Theorem 2.1). Positivity of σ2 is proved for nonconstant f under fairly general conditions, and the range of  is shown to be dense in 1⊥. A functional law of the iterated logarithm is proved when the (2+δ)th moment of f in the range of  is finite for some δ>0 (Theorem 2.7(a)). Under the additional condition of convergence in norm of the transition probability p(t, x, d y) to m(dy) as t → ∞, for each x, the above results hold when the process starts away from equilibrium (Theorems 2.6, 2.7 (b)). Applications to diffusions are discussed in some detail. © 1982 Springer-Verlag.