Generalization To The Non-Ergodic Case

Abstract

Let (X _n)_n be a Markov chain on a measurable space E with a transition kernel Q, and let β be a Banach space composed of measurable functions on E such that Q is quasi-compact on β, that sup_n>1 ||Q _n|| < +∞, and that Ker(Q - 1) ∩ β admits a basis of positive functions. Let ξ : E → ℝ be a measurable function. Under standard assumptions on the Fourier kernels Q(t)(x, dy) = e^itξ(y) Q(x, dy), we prove central limit theorems, local theorems and renewal theorems for (ξ(X _n))_n.