CLT for Stationary Normal Markov Chains via Generalized Coboundaries

Abstract

Let \(X = (X_{n})_{n\in \mathbb{Z}}\) be a stationary Markov chain with a stationary probability distribution μ on the state space of X and the transition operator Q :L 2(μ)→L 2(μ). Let f∈L 2(μ) be a function on the state space of X. The solvability in L 2(μ) of the Poisson equation \(f = g - Qg\) implies that the stationary sequence \((f(X_{n}))_{n\in \mathbb{Z}}\) can be represented in the form $$f(X_{n}) ={\bigl ( g(X_{n+1}) - (Qg)(X_{n})\bigr )} +{\bigl ( g(X_{n}) - g(X_{n+1})\bigr )} =\eta _{n} +\zeta _{n}\,(n \in \mathbb{Z}).$$ Here \(\eta = (\eta _{n})_{n\in \mathbb{Z}}\) is a stationary sequence of square integrable martingale differences, and \(\zeta = (\zeta _{n})_{n\in \mathbb{Z}}\) is an L 2-coboundary that is a difference of two consecutive elements of a stationary sequence of square integrable random variables. This representation reduces the Central Limit Theorem (CLT) question for \((f(X_{n}))_{n\in \mathbb{Z}}\) to the well-studied case of martingale differences. However, in many situations the martingale approximation as a tool in limit theorems works well, though the above martingale-coboundary representation does not hold. In particular, if the transition operator Q is normal in L 2(μ), 1 is a simple eigenvalue of Q, and the assumptions (1) \(\sigma _{f}^{2} =\int _{D}\frac{1-\vert z{\vert }^{2}} {\vert 1-z{\vert }^{2}} \rho _{f}dz < \infty \), (2) \(\lim _{n\rightarrow \infty }{n}^{-\frac{1} {2} }\vert \sum _{k=0}^{n-1}{Q}^{k}f\vert _{2} = 0\)