Stable limits of sums of bounded functions of long-memory moving averages with finite variance

Abstract

We discuss limit distributions of partial sums of bounded functions h of a long-memory moving-average process X_t= ∑_j=1 ^∞ b_j ζ_t-j with coefficients b_j decaying as j^-β, 1/2< β< 1, and independent and identically distributed innovations ζ_s whose probability tails decay as x^-α, 2< α< 4. The case of h having Appell rank k_*=2 or 3 is discussed in detail. We show that in this case and in the parameter region αβ< 2 , the partial sums process, normalized by N^1/αβ , weakly converges to an αβ-stable Lévy process, provided that the normalization dominates the corresponding k_* th-order Hermite process normalization, or that 1/αβ> 1 - (2β-1)k_*/2. A complete characterization of limit distributions of the partial sums process remains open.