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

We apply to a sequence of i.i.d. random variables a time change operator via a Poisson process that is independent of this sequence. We consider sums of independent copies of processes constructed in this way and having continuous time. Finite limit distributions of these sums coincide with the finite limit distributions of the Wiener–Ornstein–Uhlenbeck field that is the tensor product of a Brownian motion and the Ornstein–Uhlenbeck process. The transition characteristics of the limit Ornstein–Uhlenbeck process are described by Brownian bridges that are builded into the Wiener–Ornstein–Uhlenbeck field. Bibliography: 4 titles.