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Accueil du site > Séminaires > Probabilités Statistiques et réseaux de neurones > Time-varying fractionnally integrated processes with discrete and continuous argument

Vendredi 31 mars 2006 à 12 heures

Time-varying fractionnally integrated processes with discrete and continuous argument

Donatas SURGAILIS (membre de l’Académie des Sciences de Lituanie)

Abstract : Extending the works Philippe et al. (2005, 2006) on time-varying fractionally integrated operators A({\bf d}), B({\bf d}) with discrete argument depending on an arbitrary sequence {\bf d} = (d_t, t \in {\Z}) of real numbers, we introduce nonhomogenous generalizations I^{\alpha (·)} and D^{\alpha (·)} of the Liouville fractional integral and derivative operators on the real line, where \alpha (u), u\in {\R} a general function taking values in (0,1) and satisfying some regularity conditions. The proof of D^{\alpha (·)} I^{\alpha (·)}f = f relies on a surprising integral identity. We also discuss small and large scale limits of white noise integrals X_t = \int_0^t (I^{\alpha (·)} \dot B)(s) {\d}s and Y_t = \int_0^t (D^{\alpha (·)} \dot B)(s) {\d}s . In the second part of the talk we extend the results of Philippe et al. (2005, 2006) on discrete time filtered processes A({\bf d}) \veps_t and B({\bf d}) \veps_t in two directions : (1) when {\bf d} = (d_t, t \in {\Z}) is deterministic and almost periodic at +\infty and -\infty, and (2) when {\bf d} = (d_t, t \in {\Z}) is random i.i.d.

Part of the results were obtained in collaboration with Anne Philippe, Marie-Claude Viano, Paul Doukhan, Gabriel Lang, Kristina Bruzaite and Marijus Vaiciulis.

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