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