Basic Probability space, sample space concepts and order of a Stochastic Process

We all know that for a given ,  is an random variable (r.v) on some probability space, denoted by , where  is the sample space,  is the  field of subsets of  which generates the events, and  the probability defined on . Thus one can view a stochastic process,  as a family of random variables (r.v's),  on . Hence every fixed value of argument , i.e., every sample point,  depends only on  and is simply a function of one real argument, so that for each fixed value of, ,  is a real valued function defined on . Furthermore for a fixed ,  is a random variable (r.v) on  and is a function on . On the other hand for fixed, ,  is a function of , which represents a possible observation of the stochastic process, and this function  is said to be a realization or a sample function of the process. When several quantities,  are required for the complete description of the state of the system at a fixed parameter point , a generalization can be made accordingly.

For simplicity we will restrict ourselves to a single quantity  and  to be one dimension. Thus for a given value of time parameter, , of the stochastic process, , it is a simple random variable (r.v) and its probability distribution can be obtained as for any other random variable (r.v). But when  varies in a space , the information about the process  is not provided by a simple distribution for a given , but one needs the joint distribution of the basic random variables (r.v's) of the family  to get the complete information about the process. Obtaining such a joint distribution is impossible if the membership of the family is infinite in number. It then seems reasonable to assume that the behavior of the process can be obtained by studying it at discrete sets of points and accordingly a joint distribution function defined at these points seems reasonable.
So if  with  be such a discrete sets of points within , then the joint distribution of the process  at these points can be defined as , and this distribution has the simplest form when the random variables (r.v's) are independent.
The study of stochastic process does reduce to the study of simple random variable (r.v). However, in most practical cases, we are faced with the problem of assuming some sort of dependence among the random variables (r.v's). We shall restrict to the simplest type of dependence, called the first order dependence  or Markov dependence, which may be understood with the example given below.