Weak Convergence of Sample Covariance Matrices to Stochastic Integrals Via Martingale Approximations
研究了向量鞅及其差分的样本协方差矩阵在一般条件下弱收敛于矩阵随机积分，并针对严格平稳遍历序列给出了包含偏差项的类似结果，对时间序列计量经济学有参考价值。
Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral ∫ 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.