Asymptotics of partial sums of linear processes with changing memory parameter*

Abstract

We study the limit distribution of partial sums of nonstationary truncated linear process {Xt, t = 1,…, n} with long memory and changing memory parameter dt,n ∈ (0,∞). Two classes of linear processes are investigated, namely, (i) the class of FARIMA-type truncated moving averages with time-varying fractional integration parameter and (ii) the class of time-varying fractionally integrated processes introduced in [A. Philippe, D. Surgailis, and M.-C. Viano, Invariance principle for a class of nonstationary processes with long memory, C. R. Acad. Sci., Paris, Ser. I, 342:269–274, 2006; A. Philippe, D. Surgailis, and M.-C. Viano, Time-varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl., 52:651–673, 2008]. The cases of fast-changing memory parameter (dt,n = dt does not depend on n) and slowly changing memory parameter (dt,n = d(t/n) for some function d(τ), τ ∈ [0, 1]) are discussed. In the case of fast-changing memory, the limit partial sums process is a type II fractional Brownian motion (fBm) with the Hurst parameter equal to the global maximum of (dt) for class (i) processes and to the mean value of (dt) for class (ii) processes. In the case of slowly changing memory, the limit of partial sums for both classes (i) and (ii) is degenerate and “localized” at the global maximum of the memory function d(∙); however, a nondegenerate limit of the partial sums process is shown to exist when time is suitably rescaled in the vicinity of the maximum point.