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 (Wn)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 Ps=(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.