On martingale approximations

Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $\pi$ and $Q$, say $S_n=g(W_1)+\mathrm{.}\mathrm{.}\mathrm{.}+g(W_n)$, where $g$ is square integrable and has mean 0 with respect to $\pi$. If $S_n$ has the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $E(R_n^2)=o(n)$, then $g$ is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Two obvious necessary conditions are $E[E(S_n\mathrm{\mid }{W}_1{)}^2]=o(n)$ and $\lim_{n\to \infty}E(S_n^2)/n<\infty$\mathrm{.}Assumingthefirstofthese,let$\Vert g\Vert^2_+=\limsup_{n\to \infty}E(S_n^2)/n$;then$\Vert\cdot\Vert_+$ defines a pseudo norm on the subspace of $L^2(\pi )$ where it is finite. In one main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{+}$. Let $Q^{\ast }$ denote the adjoint operator to $Q$, regarded as a linear operator from $L^2(\pi )$ into itself, and consider co-isometries ($Q{Q}^{\ast }=I$), an important special case that includes shift processes. In another main result a convenient orthonormal basis for $L_0^2(\pi )$ 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.