Stochastic Processes

The following sections are included:

• Dedication
• Preface
• Contents

In this chapter the well-known and powerful harmonizability technique will be introduced as a motivation for studying several intrinsic aspects of stochastic or random processes, and random fields. It is not a mere descriptive expression to say that Fourier analytic tools are among the most important techniques in such a detailed analysis of random functions and fields. First, it is natural to consider second order processes which indicate generalizations to many classes influenced by applications and develop an extended theory as well as the consequent analysis.

As seen in Chapter 1, stochastic analysis is deepened as well as enriched, both in content and form for theory as well as applications, by a use of the methods of harmonic analysis and related integration methods. We start with an analysis of second order processes that consider together with the first and second moments, as shown in Chapter 1, implying that the usual assumption of centering at their means is somewhat restricting the general structural analysis of the (second order) processes. Also included here are the integral determination of all local functionals, and also a probability solution of the classical Riemann hypothesis. Thus the chapter contains some key aspects of the subject.

In the preceding chapter, we considered several aspects and applications of harmonizable processes, and found that in some (serious) applications we need to study further nontrivial but useful consequences of bimeasures. Their properties allow extensions of random set functions based on the (Morse-Transue or) MT-type analysis and related theory. This work also leads to key applications of filtering and signal extraction problems. Some properties of the MT-integrals were discussed in Sections 1, 2 (of Chapter 1) and they will be used here. [These are weaker than the usual Lebesgue integrals, and such (resulting) extensions will be needed and employed with details below.]

We have treated so far various properties of harmonizable random processes and of their structural analyses. But some key applications show that we need to study also various aspects of some related problems if the index is not a linear set as time axis, but is multidimensional such as the one corresponding to the space-time problems in evolution, as in Physics and elsewhere. This, therefore, takes us to considerations of random fields whose index is just a directed set, as in space-time problems mentioned above. It was also found out in early analysis that a stationary field $\left\{X_t,t\in {ℝ}^n,n>1\right\}\subset L_0^2\left(P\right)$ which satisfies an isotropy condition is the trivial one (i.e., a constant field, with probability one). Such facts lead us to analyze and establish their nontrivial structured and related properties to be used, e.g., in isotropy for harmonizable fields.

This chapter is devoted to a comprehensive treatment of harmonizable random fields on LCA groups and some results extended to hypergroups which are of interest in applications as well as in extending the subject to some new directions. We start with a brief (but abstract) account of bimeasures to use it in the ensuing analysis freely, mostly abstracted from the basic weakening of the Lebesgue analysis by M. Morse and W. Transue rendering and enabling it for the (weaker) Fréchet integrals which play a significant role in a study of (weakly) harmonizable processes. This class lies between the Riemann and Lebesgue integral analyses. Here we include a necessary outline to appreciate the general subject. Then the work proceeds with LCA groups and hyper groups.

The following sections are included:

• Introduction
• Harmonizability, Isotropy and Their Analyses
• Some Moving Averages and Sampling of Harmonizable Classes
• Multivariate Harmonizable Random Fields
• Optimum Harmonizable Filtering with Squared Loss
• Applications and Extensions of Harmonizable Fields
• Complements and Exercises
• Bibliographical Notes

The following sections are included:

• Bibliography
• Notation Index
• Author Index
• Subject Index