A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Probability. Let (Xn)n>, be a sequence of r.v.'s with E Xn = 0, E(,q=1 X1)2/n a2 > 0, SUPn,mE(Z,+n X,)2/n < oo. We prove the functional c.l.t. for (Xn) under assumptions on an(k) = sup$ I P(A n B)-P(A)P(B) | :A E a(Xi: 1 c i c m), B E a(Xi: m + k < i 1, or ac(k) = O(b k) for some b > 1, we obtain functions f#(n) such that 11 Xn 11,t = o(f6(n)) for some /3 e (2, on] is sufficient for the functional c.l.t., but the c.l.t. may fail to hold if 11 Xn II #= O(f#(n)).