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

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3.1. Definition

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

XT = {Xtj | tjTimages}

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 sequence T will be equal to images, images or images and the processes will be still denoted XT or, if we wish to be precise, Ximages, Ximages or images.

In order to be able to study correctly 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.– Any XT family with measurable mappings is called a real discrete time stochastic process:

We also say that the process is defined on the fundamental ...

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