Randomly fractionally integrated processes

Abstract

Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (dt)t∈ℤ of real numbers; if the parameter sequence is constant dt ≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)−d. They also studied partial sums limits of filtered white noise nonstationary processes A(d)et and B(d)et for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes XtA = A(d)et and XtB = B(d)et by assuming that d = (dt, t ∈ ℤ) is a random iid sequence, independent of the noise (et). In the case where the mean \(\bar d = \mathbb{E}d_0 \in \left( {0,1/2} \right)\), we show that large sample properties of XA and XB are similar to FARIMA(0, \(\bar d\), 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter \(\bar d + (1/2)\). The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(XtA) of a randomly fractionally integrated process XtA with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence dt, this reduces to the standard Hermite rank used in Dobrushin and Major [2].