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October 2008 On martingale approximations
Ou Zhao, Michael Woodroofe
Ann. Appl. Probab. 18(5): 1831-1847 (October 2008). DOI: 10.1214/07-AAP505

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(Rn2)=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)2]=o(n) and limn→∞E(Sn2)/n<∞. Assuming the first of these, let ‖g+2=lim supn→∞E(Sn2)/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 L02(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.

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Ou Zhao. Michael Woodroofe. "On martingale approximations." Ann. Appl. Probab. 18 (5) 1831 - 1847, October 2008. https://doi.org/10.1214/07-AAP505

Information

Published: October 2008
First available in Project Euclid: 30 October 2008

zbMATH: 1161.60010
MathSciNet: MR2462550
Digital Object Identifier: 10.1214/07-AAP505

Subjects:
Primary: 60F05
Secondary: 60J10

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.18 • No. 5 • October 2008
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