Limit theorems for Markov chains by the symmetrization method

详细信息    查看全文

• 作者：Christophe Cuny^a ; ^{christophe.cuny@ecp.fr" class="auth_mail" title="E-mail the corresponding author} ; Michael Lin^b ; ^{lin@math.bgu.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Forward&ndash ; backward martingale decomposition ; Sector condition ; Numerical range and Stolz region ; Normal dilation of Markov operators ; Almost sure CLT ; Poincaré ; 's inequality
• 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)$L^2(\mathbb{S},m)$, and let (W_n)_n≥0${(W_n)}_{n\ge 0}$ be the Markov chain with state space S$\mathbb{S}$ and transition probability P   on the space of trajectories 2333bc3e13e8de6f68ab2fb4d3fe" title="Click to view the MathML source">(Ω,P_m)$(\mathrm{\Omega },{\mathbb{P}}_m)$, with initial distribution m  . Following Wu and Olla we define the symmetrized operator P_s=(P+P^⁎)/2$P_s=(P+P^{⁎})/2$, and analyze the linear manifold ${\mathcal{H}}_{-1}:=\sqrt{I-P_s}{L}^2(\mathbb{S},m)$. We obtain for real f∈H₋₁$f\in {\mathcal{H}}_{-1}$ an explicit forward–backward martingale decomposition with a coboundary remainder. For such f   we also obtain some maximal inequalities for $S_n(f):={\sum }_{k=0}^{n}f(W_k)$, related to the law of iterated logarithm. We prove an almost sure central limit theorem for f∈H₋₁$f\in {\mathcal{H}}_{-1}$ when P   is normal in L²(S,m)$L^2(\mathbb{S},m)$, or when P satisfies the sector condition. We characterize the sector condition by the numerical range of P   on the complex L²(S,m)$L^2(\mathbb{S},m)$ 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 ${\mathcal{H}}_{-1}=\sqrt{I-P}{L}^2(\mathbb{S},m)$. We use our approach to prove that P   is L²$L^2$-uniformly ergodic if and only if it satisfies (the discrete) Poincaré's inequality.