3.1 Definition

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

where T called the time base is a countable set of instants. X _{t j} is the r.v. of the family considered at the instant t _j.

Ordinarily, the t _j are uniformly spread and distant from a unit of time and in the sequence T will be equal to and the processes will be still denoted X _T or, if we wish to be precise, .

In order to be able to study correctly some sets of r.v. X _j of X _T and not only the r.v. X _j 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 X _T family with measurable mappings is called a real discrete time stochastic process:

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

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

For a given ?, that is to say after the phenomenon has been performed, we obtain the values x _j = X _j ( ?).

DEFINITION. x _T = { x _j