Weak convergence to the matrix stochastic integral \(\int ^{1}_{0}B\,dB'\)
The asymptotic theory of regression with integrated processes of the ARIMA type frequently involves weak convergence to stochastic integrals of the form \(\int^{1}_{0}W dW\), 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 \(\int^{1}_{0}B dB'\), where B(r) is vector Brownian motion with a non-scalar covariance matrix. This paper studies the weak convergence of sample covariance matrices to \(\int^{1}_{0}B dB'\) under quite general conditions. The theory is applied to vector autoregressions with integrated processes.
weak convergence to matrix stochastic integrals
weak convergence of sample covariance matrices
scientific article; zbMATH DE number 4024573