On a characterization of processes by the independence of linear forms in a triangular system

Abstract

Necessary and sufficient conditions are given for a stochastic process to be either a Gaussian or an independent increments process. These conditions are based on the independence of some linear forms in a triangular system and the linearity of regression.

On stochastic processes with linear conditional moments

“…The present paper finishes, in some sense, the subject: the (LR) and (LCV) conditions imply that the process must be either a Gaussian process or a Poisson-type process. Similar theorem based on the (LR) condition and the independence of linear forms in some triangular system was recently proved for the class of Gaussian and independent increment processes (see [8]). …”