On the existence of some ARCH($\infty$) processes

Randal Douc (1) , François Roueff (2) , Philippe Soulier (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 LTCI - Laboratoire Traitement et Communication de l'Information
3 MODAL'X - Modélisation aléatoire de Paris X

A new sufficient condition for the existence of a stationary causal solution of an ARCH($\infty$) equation is provided. This condition allows to consider polynomially decaying coefficients, so that it can be applied to the so-called FIGARCH processes, whose existence is thus proved.

Mots clés: ARCH processes Fractionaly integrated processes Long memory

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Probabilités [math.PR]

Randal Douc, François Roueff, Philippe Soulier. On the existence of some ARCH($\infty$) processes. Stochastic Processes and their Applications, 2008, 118 (5), pp.755-761. ⟨10.1016/j.spa.2007.06.002⟩. ⟨hal-00113157v3⟩