ON MARTINGALE APPROXIMATIONS

• 成果类型: Article
• 署名作者: Zhao, Ou; Woodroofe, Michael
• 署名单位: Yale University; University of Michigan System; University of Michigan
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP505
• 发表日期: 2008
• 页码: 1831-1847
• 关键词: CENTRAL-LIMIT-THEOREM; weak invariance-principle; stationary-processes; additive-functionals; markov-chains; criteria

摘要

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 vertical bar W-1)(2)] = o(n) and lim(n ->infinity) E(S-n(2))/n < infinity. Assuming the first of these, let parallel to g parallel to(2)(+) = lim sup(n ->infinity) E(S-n(2))/n; then parallel to center dot parallel to(+) 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 parallel to center dot parallel to(+). 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.