Time-Varying Fractionally Integrated Processes with Finite or Infinite Variance and Nonstationary Long Memory

Abstract

Recently, Philippe et al. (C.R. Acad. Sci. Paris. Ser. I 342, 269–274, 2006; Theory Probab. Appl., 2007, to appear) introduced a new class of time-varying fractionally integrated filters A(d)x _t =∑ ^∞ _j=0 a _j (t)x _t−j , B(d)x _t =∑ ^∞ _j=0 b _j (t)x _t−j depending on arbitrary given sequence d=(d _t ,t∈ℤ) of real numbers, such that A(d)⁻¹=B(−d), B(d)⁻¹=A(−d) and such that when d _t ≡d is a constant, A(d)=B(d)=(1−L)^d is the usual fractional differencing operator. Philippe et al. studied partial sums limits of (nonstationary) filtered white noise processes X _t =B(d)ε _t and Y _t =A(d)ε _t in the case when (1) d is almost periodic having a mean value $\bar{d}\in (0,1/2)$ , or (2) d admits limits d _±=lim  _t→±∞ d _t ∈(0,1/2) at t=±∞. The present paper extends the above mentioned results of Philippe et al. into two directions. Firstly, we consider the class of time-varying processes with infinite variance, by assuming that ε _t ,t∈ℤ are iid rv’s in the domain of attraction of α-stable law (1<α≤2). Secondly, we combine the classes (1) and (2) of sequences d=(d _t ,t∈ℤ) into a single class of sequences d=(d _t ,t∈ℤ) admitting possibly different Cesaro limits $\bar{d}_{\pm}\in(0,1-(1/\alpha))$ at ±∞. We show that partial sums of X _t and Y _t converge to some α-stable self-similar processes depending on the asymptotic parameters $\bar{d}_{\pm}$ and having asymptotically stationary or asymptotically vanishing increments.