Likelihood Ratios for Processes

Abstract

As seen in the preceding work on inference theory of processes, likelihood ratios play a prominent role, particularly for the testing problems. Consequently the major part of this chapter will be devoted to finding densities for probability measures induced by broad classes of stochastic processes. These include processes with independent increments, jump Markov, and those that are infinitely divisible. Also considered for this work are diffusion types of processes and some applications. All these start with (and suggested by) Gaussian processes and so we establish results including dichotomy theorems as well as likelihood ratios for them under several different sets of conditions, together with a few stationary cases. As a motivation for (and also an interest in) the subject we start with a treatment of the important problem of (admissible) means of processes within their function space representations (and these means can be regarded as deterministic signals). The analysis presented in this chapter involves some interesting mathematical ideas, and the reader should persevere with patience, since a rich collection of problems, applications, and new directions are suggested in this work. We include many illustrations in the text, and some further results in the final complements section, often with hints.