ON A FUNCTIONAL CENTRAL LIMIT THEOREM FOR THE LINEAR PROCESS GENERATED BY ASSOCIATED RANDOM VARIABLES IN A HILBERT SPACE

Abstract

Let {<TEX>${\xi}_k,\;k\;{\in}\;{\mathbb{Z}}$</TEX>} be a strictly stationary associated sequence of H-valued random variables with <TEX>$E{\xi}_k\;=\;0$</TEX> and <TEX>$E{\parallel}{\xi}_k{\parallel}^2\;</TEX><TEX><</TEX><TEX>\;{\infty}$</TEX> and {<TEX>$a_k,\;k\;{\in}\;{\mathbb{Z}}$</TEX>} a sequence of linear operators such that <TEX>${\sum}_{j=-{\infty}}^{\infty}\;{\parallel}a_j{\parallel}_{L(H)}\;</TEX><TEX><</TEX><TEX>\;{\infty}$</TEX>. For a linear process <TEX>$X_k\;=\;{\sum}_{j=-{\infty}}^{\infty}\;a_j{\xi}_{k-j}$</TEX> we derive that {<TEX>$X_k</TEX>} fulfills the functional central limit theorem.