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

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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.

Original languageEnglish (US)
Pages (from-to)185-201
Number of pages17
JournalZeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
Volume60
Issue number2
DOIs
StatePublished - Jun 1 1982

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Mathematics(all)

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