Holderian functional central limit theorem for linear processes

Abstract

Let (Xt)t ≥ 1 be a linear process defined by Xt = ∑i=0∞ψi εt-1 where (ψi, i ≥ 0) is a sequence of real numbers and (εi , i ∈ Z) is a sequence of random variables with null expectation and variance 1. This paper provides Hölderian FCLT for (Xt)t ≥ 1 with wide class of filters. Filters with ψ(i) = l(i)/i for a slowly varying function l(i) are allowed. The weak convergence of polygonal line process build from sums of (Xt)t ≥ 1 to the standard Brownian motion W in the Hölder space (Hα), 0 < α < 1/2 - 1/τ holds provided the proper noise behavior is satisfied: E|ε1|τ < ∞, τ > 2.