Mathematics – Probability
Scientific paper
2007-08-30
Annals of Applied Probability 2008, Vol. 18, No. 5, 1831-1847
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/07-AAP505 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst
Scientific paper
10.1214/07-AAP505
Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $\pi$ and $Q$, say $S_n=g(W_1)+...+g(W_n)$, 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 $E[E(S_n|W_1)^2]=o(n)$ and $\lim_{n\to \infty}E(S_n^2)/n<\infty$. Assuming the first of these, 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 $\Vert\cdot\Vert_+$. Let $Q^*$ denote the adjoint operator to $Q$, regarded as a linear operator from $L^2(\pi)$ 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 $L_0^2(\pi)$ 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.
Woodroofe Michael
Zhao Ou
No associations
LandOfFree
On martingale approximations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On martingale approximations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On martingale approximations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-206158