Definition of Stochastic Process

Defintion 1

As already mentioned a stochastic process is a function of two parameters which are from the sample space and the parameter space respectively, i.e., , where  and , such that the general nomenclature of denoting a stochastic process is . When the realized value is observed or achieved then we say that the realization of the stochastic process has been observed and it is denoted by .

Defintion 2

A Stochastic process is a family of random (r.v.) and they are usually indexed/identifies by say , i.e., its representation is . Here one should remember that  is some index set in a way such that all the elements of  are elements of . A realization (or a sample function) of a stochastic process  is an assignment to each  of a possible value of . Here  could be finite, countable or uncountable finite. Let us illustrate this with a simple example.