Weak Convergence of Stochastic Processes Defined on Semi-infinite Time Intervals

In the standard theorems on weak convergence of stochastic processes, it is invariably assumed that the parameter set is a bounded interval. The object of this paper is to indicate that analogues of these theorems for unbounded intervals are also valid. We shall confine our attention to the results of Skorohod [l], and in particular to those results concerning his Ji topology. Let £ be a complete separable metric space with metric p. We denote by K the space of all E-valued functions x(t), 0^/< », which at every point have a limit from the left and are continuous from the right. We define on K the topology Ji: a sequence x„(f) is said to be /i-convergent to x(t) if there exists a sequence of continuous one-toone mappings \n(t) of the interval [0, ») onto itself such that for each A7>0