On martingale approximations

Abstract

Consider additive functionals of a Markov chain W_k, with stationary (marginal) distribution and transition function denoted by π and Q, say S_n=g(W₁)+⋯+g(W_n), where g is square integrable and has mean 0 with respect to π. If S_n has the form S_n=M_n+R_n, where M_n is a square integrable martingale with stationary increments and E(R_n²)=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(S_n|W₁)²]=o(n) and lim_n→∞ E(S_n²)/n<∞. Assuming the first of these, let ‖g‖₊²=lim sup_n→∞ E(S_n²)/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.