Weak convergence to the matrix stochastic integral [integral operator]01 B dB'

The asymptotic theory of regression with integrated processes of the ARIMA type frequently involves weak convergence to stochastic integrals of the form [integral operator]01 W dW, 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 operator]01 B dB', where B(r) is vector Brownian motion with a non-scalar covariance matrix. This paper studies the weak convergence of sample covariance matrices to [integral operator]01 B dB' under quite general conditions. The theory is applied to vector autoregressions with integrated processes.