Representation Of Stochastic-Processes Without Loss Of Information

This note considers the problem of representing a continuous parameter stochastic process $\{ {x( {t,\omega } ),t \in T} \}$ by a sequence of random variables $\{ {\xi _n ( \omega )} \}_n $ without loss of information. The random variables are obtained in terms of linear integral operations on the sample paths of the process. A class of stochastic processes for which such a representation is possible is given in [2]. It is shown here that two new broad classes of stochastic processes admit such a representation. The first is the class of second order, continuous in probability stochastic processes. The second is the class of weakly continuous stochastic processes. It is further shown that these three classes are distinct.