On martingale approximations

Abstract

Consider additive functionals of a Markov chain Wk, with stationary (marginal) distribution and transition function denoted by π and Q, say Sn=g(W1)+⋯+g(Wn), where g is square integrable and has mean 0 with respect to π. 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(π) 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(π) 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(π) 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.