Weakly dependent chains with infinite memory

Paul Doukhan (1, 2) , Olivier Wintenberger (2, 3)

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

We prove the existence of a weakly dependent strictly stationary solution of the equation $ X_t=F(X_{t-1},X_{t-2},X_{t-3},\ldots;\xi_t)$ called {\em chain with infinite memory}. Here the {\em innovations} $\xi_t$ constitute an independent and identically distributed sequence of random variables. The function $F$ takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments and the rate of decay of the Lipschitz coefficients of the function $F$. With the help of the weak dependence properties, we derive Strong Laws of Large Number, a Central Limit Theorem and a Strong Invariance Principle.

Domains: Probability [math.PR]; Statistics [math.ST]; Statistics Theory [stat.TH]

Paul Doukhan, Olivier Wintenberger. Weakly dependent chains with infinite memory. Stochastic Processes and their Applications, 2008, 118 (11), pp.1997-2013. ⟨hal-00199890⟩