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

Peter Phillips

Econometric Theory, 1988, vol. 4, issue 3, 528-533

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.

Date: 1988
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Working Paper: Weak Convergence of Sample Covariance Matrices to Stochastic Integrals via Martingale Approximations (1987)
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