# Weakly dependent chains with infinite memory

Abstract : 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.
Document type :
Journal articles
Domain :

Cited literature [32 references]

https://hal-paris1.archives-ouvertes.fr/hal-00199890
Contributor : Olivier Wintenberger <>
Submitted on : Wednesday, December 19, 2007 - 5:42:20 PM
Last modification on : Thursday, April 22, 2021 - 1:20:03 PM
Long-term archiving on: : Monday, April 12, 2010 - 8:34:54 AM

### Files

DoukhanWintenberger9.pdf
Publisher files allowed on an open archive

### Identifiers

• HAL Id : hal-00199890, version 1
• ARXIV : 0712.3231

### Citation

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

Record views