Linear Problems in P - Th Order and Stable Processes

This work extends to processes with finite moments or order p, 1 < P < 2 and to symmetric a-stable processes, 1 < a < 2, some of the basic linear theory known for processes with finite second moments (p=2) and for Gaussian processes (a=2). Here the IIcovariationll plays a role analogous to the covariance. Specifically, stochastic integrals of two types are introduced and studied for p-th order processes and in particular for symmetric stable processes. Regression estimates and linear estimates on certain symmetric stable processes are evaluated, including regression and linear filtering of signal in noise. Also, for certain symmetric stable inputs, the identification of a linear system from the input covariation and the input-output cross co-variation is considered, and the way the distribution of the output depends on the linear system is studied.