On Martingale Approximations

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(R2 n) = 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)] = o(n) and limn→∞E(S n)/n < ∞. Assuming the first of these, let ‖g‖+ = lim supn→∞E(S n)/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 L0(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.