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Weak Convergence of Sample Covariance Matrices to Stochastic Integrals Via Martingale Approximations

Published online by Cambridge University Press:  18 October 2010

P.C.B. Phillips
P.C.B. Phillips*
Affiliation:
Cowles Foundation, Yale University
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Abstract

Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral ∫01BdB′, where B is vector Brownian motion. For strictly stationary and ergodic sequences, rather than martingale differences, a similar result obtains. In this case, the limit is ∫01BdB′ + Λ and involves a constant matrix Λ of bias terms whose magnitude depends on the serial correlation properties of the sequence. This note gives a simple proof of the result using martingale approximations.

Type
Brief Report
Information
Econometric Theory , Volume 4 , Issue 3 , December 1988 , pp. 528 - 533
Copyright
Copyright © Cambridge University Press 1988 

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References

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