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.
Phillips, Peter C.B., "Weak Convergence of Sample Covariance Matrices to Stochastic Integrals via Martingale Approximations" (1987). Cowles Foundation Discussion Papers. 1089.