Gaussian noise related to generalised Ehrenfest model

In this article we consider the generalization of Ehrenfest model, where at each moment of time not 1 but some k of n particles go from one box to another. We describe this process by a sequence of Bernoulli random vectors. We define related Bernoulli noise on a set of continuous functions for different times, and prove that it converges to Ornstein-Uhlenbeck sequence of Gaussian white noises when number of particles tends to infinity.