Randomly fractionally integrated processes (original) (raw)

2007, Lithuanian Mathematical Journal

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 = (d t) t∈Z of real numbers; if the parameter sequence is constant d t ≡ 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)ε t and B(d)ε t for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X A t = A(d)ε t and X B t = B(d)ε t by assuming that d = (d t , t ∈ Z) is a random iid sequence, independent of the noise (ε t). In the case where the meand = Ed 0 ∈ (0, 1/2), we show that large sample properties of X A and X B are similar to FARIMA(0,d, 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameterd + (1/2). The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(X A t) of a randomly fractionally integrated process X A t 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 d t , this reduces to the standard Hermite rank used in Dobrushin and Major [2].

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