Empirical spectral processes and their applications to time series analysis (Q1109413)

Let X(t) be an r-dimensional strictly stationary Gaussian time series with a matrix \(F=(F_{ab})\) of spectral distribution functions (SDF's). Let \(F^{(N)}\) be the matrix of empirical SDF's. Let \[ E_ N(g)=(N^{1/2}\int^{\pi}_{- \pi}g_{ab}(\lambda)\{dF_{ab}^{(N)}(\lambda)- dF_{ab}(\lambda)\})_{a,b=1,...,r}. \] The author proves weak convergence and equicontinuity of \(E_ N(g)\) for some classes of functions used as index sets for empirical processes. Some applications of general results to the theory of time series are indicated.