On martingale approximations

Consider additive functionals of a Markov chain Wk, with stationary (marginal) distribution and transition function denoted by . and Q, say Sn=g(W1)+.+g(Wn), where g is square integrable and has mean 0 with respect to .. If Sn has the form Sn=Mn+Rn, where Mn is a square integrable martingale with stationary increments and E(Rn2)=o(n), then g is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Two obvious necessary conditions are E[E(Sn|W1)2]=o(n) and limn...E(Sn2)/n<.. Assuming the first of these, let .g.+2=lim.supn...E(Sn2)/n; then ...+ defines a pseudo norm on the subspace of L2(.) where it is finite. In one main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of ...+. Let Q* denote the adjoint operator to Q, regarded as a linear operator from L2(.) into itself, and consider co-isometries (QQ*=I), an important special case that includes shift processes. In another main result a convenient orthonormal basis for L02(.) is identified along with a simple necessary and sufficient condition for the existence of a martingale approximation in terms of the coefficients of the expansion of g with respect to this basis.