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},...;\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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2007
- DOI:
- 10.48550/arXiv.0712.3231
- arXiv:
- arXiv:0712.3231
- Bibcode:
- 2007arXiv0712.3231D
- Keywords:
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- Mathematics - Probability;
- Mathematics - Statistics
- E-Print:
- Stochastic Processes and their Applications (2008) accept\'e