caa a22 4500 463177002 CHVBK 20180406164828.0 cr unu---uuuuu 170326e20070101xx s 000 0 eng 10.1007/s10986-007-0001-2 doi (NATIONALLICENCE)springer-10.1007/s10986-007-0001-2 Randomly fractionally integrated processes [Elektronische Daten] [P. Doukhan, G. Lang, D. Surgailis] 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 t A = A(d)ε t and X t B = 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 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 in Dobrushin and Major [2]. Springer Science+Business Media, Inc., 2007 fractional derivatives and integrals nationallicence self-similar processes nationallicence central limit theorem nationallicence noncentral limit theorems nationallicence FARIMA process nationallicence Doukhan P. Laboratoire de Statistique du CREST, Timbre J340, 3, avenue Pierre Larousse, 92240, Malakoff, France aut Lang G. Laboratoire GRESE, ENGREF, 19 av. du Maine, 75732, Paris Cedex 15, France aut Surgailis D. Institute of Mathematics and Informatics, Akademijos 4, LT-08663, Vilnius, Lithuania aut Lithuanian Mathematical Journal Kluwer Academic Publishers-Consultants Bureau 47/1(2007-01-01), 1-23 0363-1672 47:1<1 2007 47 10986 https://doi.org/10.1007/s10986-007-0001-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10986-007-0001-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Doukhan P. Laboratoire de Statistique du CREST, Timbre J340, 3, avenue Pierre Larousse, 92240, Malakoff, France aut NATIONALLICENCE 700 1- Lang G. Laboratoire GRESE, ENGREF, 19 av. du Maine, 75732, Paris Cedex 15, France aut NATIONALLICENCE 700 1- Surgailis D. Institute of Mathematics and Informatics, Akademijos 4, LT-08663, Vilnius, Lithuania aut NATIONALLICENCE 773 0- Lithuanian Mathematical Journal Kluwer Academic Publishers-Consultants Bureau 47/1(2007-01-01), 1-23 0363-1672 47:1<1 2007 47 10986 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer