A central limit theorem for reversible processes with non-linear growth of variance

الأصل · EN

Kipnis and Varadhan showed that for an additive functional, Sₙ say, of a reversible Markov chain the condition E(Sₙ²)/n → κ∈ (0,∞) implies the convergence of the conditional distribution of Sₙ/√E(Sₙ²), given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, E(Sₙ²) = nℓ(n), where ℓ is a slowly varying function. It is shown by example that the conditional distribution of Sₙ/√E(Sₙ²) need not converge to the standard normal distribution in this case; and sufficient conditions for convergence to a (possibly non-standard) normal distribution are developed.