### Abstract

Let X_{t}:t≧0 be an ergodic stationary Markov process on a state space S. If Â is its infinitesimal generator on L^{2}(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 σ^{2} =-2〈f, g〉=-2〈Âg, g〉 where g is some element in the domain of Â such that Âg=f (Theorem 2.1). Positivity of σ^{2} is proved for nonconstant f under fairly general conditions, and the range of Â 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 Â 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 language | English (US) |
---|---|

Pages (from-to) | 185-201 |

Number of pages | 17 |

Journal | Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete |

Volume | 60 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1982 |

### Fingerprint

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Mathematics(all)