Limit theorems for Markov chains by the symmetrization method

Let P be a Markov operator with invariant probability m, ergodic on L2(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 (Ω,Pm), 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:=I-√PsL²(S,m). We obtain for real f∈H-1 an explicit forward-backward martingale decomposition with a coboundary remainder. For such f we also obtain some maximal inequalities for Sn(f):=∑k=0nf(Wk), related to the law of iterated logarithm. We prove an almost sure central limit theorem for f∈H-1 when P is normal in L2(S,m), or when P satisfies the sector condition. We characterize the sector condition by the numerical range of P on the complex L2(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=√I-PL²(S,m). We use our approach to prove that P is L²-uniformly ergodic if and only if it satisfies (the discrete) Poincaré's inequality.