In: Statistics and Probability
Ans-
The class of all stochastic processes is too large to work with in practice. We consider only the subclass of stationary processes.
{Xt} is said to be
completely stationary if, for all n 1, for any
t1,t2, . . . ,tn T,
and for any such that
t1+,t2+
, . . .
,tn+
T are also contained in the index set,
the joint cdf of {Xt1 ,
Xt2 , . . . , Xtn } is the same as that of
{Xt1+ ,
Xt2+
, . . . ,
Xtn+
}
i.e.,
Ft1,t2,...,tn (a1, a2, . . . , an)
= Ft1+,t2+
,...,tn+
(a1,
a2, . . . , an),
so that the probabilistic structure of a completely stationary process is invariant under a shift in time