Functional Central Limit Theorems for Some Semi-Stationary Processes

In [6], Serfozo introduced a class of stochastic processes which he called semi-stationary processes, and discussed elementary properties of these processes. Moreover, he presented ergodic theorems and functional central limit theorems with the uniformly strong mixing condition for them. These processes are a generalization of strictly stationary processes. Hereafter stationary means strictly stationary. The object of this paper is to prove the functional central limit theorems for the semi-stationary processes with the strong mixing condition. These results contain as special cases the corresponding theorems for the stationary processes (Theorem 1, Theorem 2 in [4])