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

Authors

Peter C.B. Phillips

Document Type

Discussion Paper

Publication Date

7-1-1987

CFDP Number

846

CFDP Pages

9

Abstract

Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral from zero to one of ∫ 0 1 BdB ’, 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 ∫ 0 1 BdB ’ + 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.

Recommended Citation

Phillips, Peter C.B., "Weak Convergence of Sample Covariance Matrices to Stochastic Integrals via Martingale Approximations" (1987). Cowles Foundation Discussion Papers. 1089.
https://elischolar.library.yale.edu/cowles-discussion-paper-series/1089