Some Invariance Principles for Functionals of a Markov Chain

The new results are (C), (D) and (F) below. Let X₀, X₁, ⋯ be a Markov chain with countable state space I and stationary transitions. Suppose I is a positive recurrent class, with stationary probability vector p. Let f be a real-valued function on I. Fix a reference state s ε I, and let 0 t₁ < t₂ < ⋯ be the times n at which Xₙ = s. Let Yⱼ = ∑ (Xₙ):tⱼ n < tj + 1\ and Uⱼ = ∑ \|f(Xₙ)|:tⱼ n < tj + 1\. Let Vₘ = ∑ᵐj = 1 Yⱼ and Sₙ = ∑ⁿj = 0f(Xⱼ). For (C) and (D) below, assume (A) ∑i ε I pᵢf(i) = 0; and (B) U²ⱼ has finite expectation. Then: (C) Theorem. n-1/2 max \|Sⱼ - Vjpₛ|: 1 j n\ → 0 in probability; and (D) Theorem. (n log log n)-1/2 max \|Sⱼ - Vjpₛ|: 1 j n\ → 0 almost everywhere. For (F), do not assume (A) and (B), but assume (E) Yⱼ differs from 0 with positive probability. Let vₘ (respectively, sₙ) be 1 or 0 according as Vₘ (respectively, Sₙ) is positive or non-positive. Then (F) Theorem. n⁻¹ ∑ ⱼ : 1 j n\ - p⁻¹ₛn⁻¹ ∑ ⱼ : 1 j npₛ\ → 0 almost everywhere. I do not believe the convergence in (C) is a.e., but have no counter-example.