Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem

Let (X-i) be a stationary and ergodic Markov chain with kernel Q and f an L-2 function on its state space. If Q is a normal operator and f = (I - Q)(1/2)g (which is equivalent to the convergence of Sigma(infinity)(n=1) Sigma(n-1)(k=0) Q(k) f/n(3/2) in L-2), we have the central limit theorem [cf. (Derriennic and Lin in C. R. Acad. Sci. Paris, Ser. I 323: 1053-1057, 1996; Gordin and Lifsic in Third Vilnius conference on probability and statistics, vol. 1, pp. 147-148, 1981)]. Without assuming normality of Q, the CLT implied by the convergence of Sigma(infinity)(n=1) parallel to Sigma(n-1)(k=0) Q(k) f parallel to(2)/n(3/2), in particular by parallel to Sigma(n-1)(k=0) Q(k) f parallel to(2) = o(root n/log(q) n), q > 1 by Maxwell and Woodroofe (Ann. Probab. 28: 713-724, 2000) and Wu and Woodroofe (Ann. Probab. 32: 1674-1690, 2004), respectively. We show that if Q is not normal and f is an element of (I - Q)L-1/2(2), or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to Sigma(infinity)(n=1) c(n) parallel to Sigma(n-1)(k=0) Q(k) f parallel to(2)/n(3/2) < infinity for some sequence c(n) SE arrow 0, or by parallel to Sigma(n-1)(k=0) Q(k) f parallel to(2) = 0(root n/log n), the CLT need not hold.