A note on the normalizing sequences for sums of linear processes in the case of negative memory

Abstract

We consider the partial-sum process ${S}_n(t)={\sum}_{k=0}^{\left\lfloor nt\right\rfloor }{X}_k$ of linear processes ${X}_n={\sum}_{i=0}^{\infty }{c}_i{\upxi}_{n-i}$ with independent identically distributed innovations {ξ _i } belonging to the domain of attraction of α-stable law (0 < α ≤ 2). If |c _k | = k ^−γ , k ∈ ℕ , γ > max(1, 1/α), and ${\sum}_{k=0}^{\infty}\kern0.5em ck=0$ (the case of negative memory for the stationary sequence {X _n }), then it is known that the normalizing sequence of S _n (1) can grow as n ^1/α−γ+1 or remain bounded if the signs of the coefficients are constant or alternate, respectively. It is of interest to know whether it is possible, given ⋋ ∈ (0, 1/α − γ + 1), to change the signs of c _k so that the rate of growth of the normalizing sequence would be n ^⋋ . In this paper, we give the positive answer: we propose a way of choosing the signs and investigate the finite-dimensional convergence of appropriately normalized S _n (t) to linear fractional Lévy motion.