Weak law of large numbers for almost periodically correlated processes

Abstract

This note contains two simple observations concerning the weak law of large numbers for almost periodically correlated processes. Let H be a complex Hilbert space with the inner product (., .), and let R denote the set of real numbers. Any Borel measurable function X: R H will be referred to as a stochastic process. A stochastic process X is said to be almost periodically correlated (APC), if its correlation function R(s,u) = (X(s),X(u)) is bounded and uniformly continuous in s,u, and for each t E R the function B(t, u) = R(t + u, u), u E R, is uniformly almost periodic in u [3]. If X is APC, then for every A, t E R the limit (1) aA(t) = lim JB(t, u) exp (-iAu)du exists, the set A = {A E R: a,(t) =A 0 for some t} is countable, and for every A E A there is a complex measure. rA (sometimes called the spectral measure corresponding to al) such that a\(t) = Judo exp (itx)rF(dx) (see e.g. [4]). An APC process X is called almost periodically unitary (APU) if there is a continuous unitary group U(t), t E R, and an H-valued uniformly almost periodic function f such that X(t) = U(t)f(t), t E R [5]. An APC process X is called uniformly almost periodically correlated (UAPC) if the function u ) B(., u) is uniformly almost periodic from R to C(R), the Banach space of bounded continuous functions on R equipped with the sup-norm [2]. We will say that a process X satisfies the weak law of large numbers (WLLN) if the limit (2) lim X(t)dt exists in the norm topology of H. The problem of existence of the limit (2) for APC processes was raised in [1], where among other results it was proved that if X is APC and ZAEA-{0} A-2 is finite, then X satisfies WLLN. The purpose of this note is to show that for APC processes WLLN holds when the set A {0} is separated from zero. This result Received by the editors November 15, 1994. 1991 Mathematics Subject Classification. Primary 60G12, 60F05.