Limiting distributions of nonlinear vector functions of stationary Gaussian processes
Let \((X_ m,Y_ m)\), \(m\in Z\), be a sequence of stationary Gaussian vectors. Let H(x) and K(x) be two real functions and define \[ Z^ n_ H=A_ n^{-1}\sum^{n-1}_{m=0}H(X_ m)\text{ and } Z^ n_ K=B_ n^{-1}\sum^{n-1}_{m=0}K(Y_ m), \] where \(A_ n\) and \(B_ n\) are some constants. The purpose of this paper is to study the joint limiting distribution of \((Z^ n_ H,Z^ n_ K)\). Moreover it is shown that \(Z^ n_ H\) and \(Z^ n_ K\) are asymptotically independent in various cases. Some application of this to the limiting distribution for a certain class of nonlinear infinite-coordinated functions of a Gaussian process is also examined.
scientific article; zbMATH DE number 4172042