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Randomly fractionally integrated processes. (English) Zbl 1144.60026

Lith. Math. J. 47, No. 1, 1-23 (2007); and Liet. Mat. Rink. 47, No. 1, 3-28 (2007).
Authors’ abstract: A. Philippe, D. Surgailis, and M.-C. Viano [C. R. Acad. Sci. Paris, Ser. 1. 342, 269–274 (2006; Zbl 1086.60506); in: Dependence in probability and statistics. New York, NY: Springer. Lect. Notes Statistics 187, 159–194 (2006; Zbl 1332.62339)] introduced two distinct time-varying mutually invertible fractionally integrated filters \(A({\mathbf d}), B ({\mathbf d})\) depending on an arbitrary sequence \({\mathbf d} = (d_{t})_{t \in \mathbb Z}\) of real numbers; if the parameter sequence is constant \(d_{t} \equiv d\), then both filters \(A({\mathbf d})\) and \(B ({\mathbf d})\) reduce to the usual fractional integration operator \((1 - L)^{- d}\). They also studied partial sums limits of filtered white noise nonstationary processes \(A({\mathbf d}) \epsilon_{t}\) and \(B ({\mathbf d}) \epsilon_{t}\) for certain classes of deterministic sequences \({\mathbf d}\). The present paper discusses the randomly fractionally integrated stationary processes \(X_{t}^{A}= A ({\mathbf d}) \epsilon_{t}\) and \(X_{t}^{B}= B ({\mathbf d}) \epsilon_{t}\) by assuming that \({\mathbf d} = (d_{t}, t \in \mathbb Z)\) is a random iid sequence, independent of the noise \((\epsilon_{t})\). 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 \(X^{A}\) and \(X^{B}\) 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 \(h (X_{t}^{A})\) of a randomly fractionally integrated process \(X_{t}^{A}\) 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 by R. L. Dobrushin and P. Major [Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, No. 1, 27–52 (1979; Zbl 0397.60034)].

MSC:

60G10 Stationary stochastic processes
60F05 Central limit and other weak theorems
60G18 Self-similar stochastic processes
60G12 General second-order stochastic processes
60G15 Gaussian processes
60G35 Signal detection and filtering (aspects of stochastic processes)
62M20 Inference from stochastic processes and prediction
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References:

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