Holderian functional central limit theorem for linear processes

Abstract

Let (X_t)_{t ≥ 1} be a linear process defined by X_t =  ∑_i=0∞ψ_i ε_t-1 where (ψ_i, i ≥ 0) is a sequence of  real numbers and (ε_i , i ∈ Z) is a sequence of random variables with null expectation and variance 1. This paper provides Hölderian FCLT for (X_t)_{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 (X_t)_{t ≥ 1} to the standard Brownian motion W in the Hölder space (H_α), 0 < α < 1/2 - 1/τ holds provided the proper noise behavior is satisfied: E|ε₁|^τ < ∞, τ > 2.