2009/12/15 by Volný, Dalibor
#60F05 #60G10 #60G42 #60J05 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.0912.2864
Let (Xi) be a stationary and ergodic Markov chain with kernel Q, f an L2 function on its state space. If Q is a normal operator and f = (I-Q)1/2g (which is equivalent to the convergence of ∑n=1^∞ \frac∑k=0n-1Qkfn3/2 in L2), we have the central limit theorem (cf\. \citeD-L 1, \citeG-L 2). Without assuming normality of Q, the CLT is implied by the convergence of ∑n=1^∞ \frac‖∑k=0n-1Qkf‖2n3/2, in particular by ‖∑k=0n-1Qkf‖2 = o(√ n/logq n), q>1 by \citeM-Wu and \citeWu-Wo respectively. We shall show that if Q is not normal and f∈ (I-Q)1/2 L2, or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to ∑n=1^∞ cn\frac‖∑k=0n-1Qkf‖2n3/2