刊名:Journal of Mathematical Analysis and Applications
出版年:2016
出版时间:1 February 2016
年:2016
卷:434
期:1
页码:52-83
全文大小:705 K
文摘
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 2333bc3e13e8de6f68ab2fb4d3fe" title="Click to view the MathML source">(Ω,Pm), with initial distribution m . Following Wu and Olla we define the symmetrized operator Ps=(P+P⁎)/2, and analyze the linear manifold . 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 , 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 . We use our approach to prove that P is L2-uniformly ergodic if and only if it satisfies (the discrete) Poincaré's inequality.