Stochastic Processes of Second Order with Values in Banach Spaces

Abstract

In [6] Kolmogorov studied second order stationary one-dimensional processes as curves in Hilbert space. The idea again occured in the paper [4] of Cramer for non-stationary processes. By generalizing the concept of Hilbert space in such a way that the inner product takes values which are no longer scalars but elements of more general topological space, it is possible to extend the above model to infinite-dimensional stochastic processes of second order with values in a suitable abstract space. This idea was studied in [1] for stationary processes, where a pseudo-hilbertian space (LVH-space which first appeared in the paper [7] of Loynes) had been used as the generalized Hilbert space. Independently Saworotnow in [10] proposed a Hilbert module as such a space. We remark that the situation in [1] is in contrast with another generalization of second order stationarity given in [8], where LVH-space valued stationary processes are studied.