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On Martingale Approximations 1
| Content Provider | Semantic Scholar |
|---|---|
| Author | Woodroofe, Michael |
| Abstract | Consider additive functionals of a Markov chain W k , 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. Two obvious necessary conditions are E[E(Sn|W1) 2 ] = o(n) and limn→∞ E(S 2 n)/n < ∞. Assuming the first of these, let g 2 + = lim sup n→∞ E(S 2 n)/n; then · + defines a pseudo norm on the sub-space of L 2 (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 L 2 (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 L 2 0 (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. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/0708.4183v2.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Approximation Coefficient Expanded memory Marginal model Markov chain Pseudo brand of pseudoephedrine Radon Stationary process Tencent QQ Utility functions on indivisible goods |
| Content Type | Text |
| Resource Type | Article |
