A ug 2 00 7 On Martingale Approximations ∗
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چکیده
Abstract 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(R 2 n) = o(n), then g is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Let Q denote the adjoint operator to Q, regarded as a linear operator from L(π) into itself, and consider co-isometries (QQ = I), an important special case that includes shift processes. In one 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. Two obvious necessary conditions for a martingale approximation 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 L(π) where it is finite. In another main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of ‖ · ‖+.
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تاریخ انتشار 2008