A Functional Limit Theorem for Dependent Sequences with Infinite Variance Stable Limits

Johan Segers (U.C.Louvain, Belgique)
vendredi 18 juin 2010

Résumé : Under an appropriate regular variation condition, the affinely
normalized partial sums of a sequence of independent and identically
distributed random variables converges weakly to a non-Gaussian stable
random variable. A functional version of this is known to be true as
well, the limit process being a stable Lévy process. The main result
in the paper is that for a stationary, regularly varying sequence for
which clusters of high-threshold excesses can be broken down into
asymptotically independent blocks, the properly centered partial sum
process still converges to a stable Lévy process. Due to clustering,
the Lévy triple of the limit process can be different from the one in
the independent case. The convergence takes place in the space of
càdlàg functions endowed with Skorohod’s M_1 topology, the more usual
J_1 topology being inappropriate as the partial sum processes may
exhibit rapid successions of jumps within temporal clusters of large
values, collapsing in the limit to a single jump. The result rests on
a new limit theorem for point processes which is of independent
interest. The theory is applied to moving average processes, squared
GARCH(1,1) processes, and stochastic volatility models.

Travail en collaboration avec Bojan BASRAK (University of Zagreb) et
Danijel KRIZMANIC (University of Rijeka).

Cet exposé se tiendra à 11h00 en salle C20-13, 20ème étage, Université
Paris 1, Centre Pierre Mendes-France, 90 rue de Tolbiac, 75013 Paris
(métro : Olympiades).