Weak Convergence to the Matrix Stochastic Integral BdB

The asymptotic theory of regression with integrated processes of the ARIMA type frequently involves weak convergence to stochastic integrals of the form integral_{0}^{1}WdW, where W(r) is standard Brownian motion. In multiple regressions and vector autoregressions with vector ARIMA processes the theory involves weak convergence to matrix stochastic integrals of the form integral_{0}^{1}BdB', where B(r) is vector Brownian motion with non scalar covariance matrix. This paper studies the weak convergence of sample covariance matrices to integral_{0}^{1}BdB' under quite general conditions. The theory is applied to vector autoregressions with integrated processes.