Randomly fractionally integrated processes

• Autores: P. Doukhan, G. Lang, Donatas Surgailis
• Localización: Lithuanian mathematical journal, ISSN 0363-1672, Vol. 47, Nº. 1, 2007, págs. 1-23
• Idioma: inglés
• DOI: 10.1007/s10986-007-0001-2
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• Resumen

  • 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)e t and B(d)e t for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X t A = A(d)e t and X t B = B(d)e t by assuming that d = (d t , t ? Z) is a random iid sequence, independent of the noise (e t ). In the case where the mean , we show that large sample properties of X A and X B are similar to FARIMA(0, , 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter . 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].