Weakly dependent chains with infinite memory

2007/12/19 by Paul Doukhan, Olivier Wintenberger, Doukhan, Paul +1 · 1 citation
Mathematics · Economics, Econometrics and Finance · #Stochastic processes and statistical mechanics #Stochastic processes and financial applications #Markov Chains and Monte Carlo Methods

paper · doi:10.48550/arxiv.0712.3231

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,...;ξ_t) called \em chain with infinite memory. Here the \em innovations ξ_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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