Limit theorems for Markov chains by the symmetrization method
Résumé
Let P be a Markov operator with invariant probability m, ergodic on L-2(S, m), and let (W-n)(n >= 0) be the Markov chain with state space S and transition probability P on the space of trajectories (Omega, P-m), with initial distribution m. Following Wu and Olla we define the symmetrized operator P-s = (P P*)/2, and analyze the linear manifold H-1. = root I-PsL2(S, m). We obtain for real f is an element of H-1 an explicit forward backward martingale decomposition with a coboundary remainder. For such f we also obtain some maximal inequalities for S-n(f) = Sigma(n)(k=0) f (W-k), related to the law of iterated logarithm. We prove an almost sure central limit theorem for f is an element of H-1. when P is normal in L-2(S, m), or when P satisfies the sector condition. We characterize the sector condition by the numerical range of P on the complex L-2(S, m) being in a sector with vertex at 1. We then show that if P has a real normal dilation which satisfies the sector condition, then H-1 = root I-PL2(S, m). We use our approach to prove that P is L-2-uniformly ergodic if and only if it satisfies (the discrete) Poincare's inequality. (C) 2015 Elsevier Inc. All rights reserved.