Limit theorems for Markov chains by the symmetrization method

Christophe Cuny, Michael Lin

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)52-83
Number of pages32
JournalJournal of Mathematical Analysis and Applications
Volume434
Issue number1
DOIs
StatePublished - 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

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