Weakly dependent chains with infinite memory

TL;DR

This paper establishes the existence of weakly dependent chains with infinite memory, explores their moment properties, and derives key probabilistic limit theorems using weak dependence techniques.

Contribution

It proves the existence of stationary solutions for chains with infinite memory under Lipschitz conditions and analyzes their probabilistic properties.

Findings

Existence of weakly dependent stationary solutions

Conditions linking moments and Lipschitz decay rates

Derivation of SLLN, CLT, and invariance principles

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.