## Abstract

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-√PsL^{2}(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-PL^{2}(S,m). We use our approach to prove that P is L^{2}-uniformly ergodic if and only if it satisfies (the discrete) Poincaré's inequality.

Original language | English |
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Pages (from-to) | 52-83 |

Number of pages | 32 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 434 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 2016 |

## Keywords

- Almost sure CLT
- Forward-backward martingale decomposition
- Normal dilation of Markov operators
- Numerical range and Stolz region
- Poincaré's inequality
- Sector condition

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics