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

• R. N. Bhattacharya
Article

Summary

Let Xt∶t≧0 be an ergodic stationary Markov process on a state space S. If Â 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 $$\hat A,n^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \int\limits_0^{nt} {f(X_s ){\text{ }}ds{\text{ }}(t\underline{\underline > } } 0)$$ 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 Â such that Âg=f (Theorem 2.1). Positivity of σ2 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.

Keywords

State Space Probability Measure Limit Theorem Variance Parameter Markov Process
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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