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Discrete Stochastic Processes and Optimal Filtering, 2nd Edition by Roger Ceschi, Jean-Claude Bertein

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Chapter 3

Introduction to Discrete Time Processes

3.1. Definition

A discrete time process is a family of r.v.

image

where T, called the time base, is a countable set of instants. Xtj is the r.v. of the family considered at the instant tj.

Ordinarily, the tj are uniformly spread and distant from a unit of time and in the following T will be equal to image and the processes will still be denoted XT or, if we wish to be precise, image.

In order to be able to correctly study some sets of r.v. Xj of XT and not only the r.v. Xj individually, it is in our interests to consider the latter as being definite mappings on the same set and this leads us to an exact definition.

DEFINITION.– We call a discrete time stochastic process any family XT of measurable mapping:

image

We also say that the process is defined on the fundamental space (Ω,a).

In general a process XT is associated with a real phenomenon, that is to say that the Xj represent (random) physical, biological, etc. values, for example, the intensity of electromagnetic noise coming from a certain star.

For a given ω, that is to say after the phenomenon has ...

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