Poissonian subordinators, the Wiener–Ornstein–Uhlenbeck field, and a relation between the Ornstein–Uhlenbeck processes and the Brownian bridges

To a sequence of i.i.d. random variables we apply the time change operator via the independent of this sequence Poisson process. We consider the sums of the independent copies of a such kind constructed processes having the continuous time. The finite dimensional distributions of the limits of these sums coincide with the finite dimensional distributions of the Wiener–Ornstein–Uhlenbeck field which is the tensor product of the Brownian motion and the Ornstein–Uhlenbeck process. The transition characteristics of the limiting process are described by the Brownian bridges which are built into the Wiener–Ornstein–Uhlenbeck field. Bibl. 4 titles.