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

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

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