A LARCH($\infty$) Vector Valued Process

Paul Doukhan (1, 2, 3) , Gilles Teyssière (2, 3) , Pablo Winant (1)

1 CREST - Centre de Recherche en Économie et Statistique
2 SAMOS - Statistique Appliquée et MOdélisation Stochastique
3 CES - Centre d'économie de la Sorbonne

We introduce a vector version of the ARCH($\infty$) equation yielding a simple approach to various models like bilinear or GARCH models. To this aim we provide an explicit chaotic expansion of a solution for this LARCH equation, and show the uniqueness of this solution under reasonable conditions. Independent or $N$-Markov approximations of this process allow to simulate their trajectory or to derive bounds for their weak dependence coefficients as defined by Doukhan and Louhichi (1999). We finally consider a long range dependent version of this model; in this case we provide an existence and uniqueness result.

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

Paul Doukhan, Gilles Teyssière, Pablo Winant. A LARCH($\infty$) Vector Valued Process. Patrice Bertail, Paul Doukhan, Philippe Soulier. Dependence in Probability and Statistics, Springer, pp.26, 2006, Lecture Notes in Statistics. ⟨hal-00141714⟩