Weakly dependent chains with infinite memory

Abstract

We prove the existence of a weakly dependent strictly stationary solution of the equation Xt = F (Xt−1, Xt−2, Xt−3, . . . ; ξt) called chain with infinite memory. Here the innovations ξt 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.2000 Mathematics Subject Classification. Primary 62M10; Secondary 91B62, 60K35, 60K99, 60F05, 60F99.

Asymptotic theory for stationary processes

“…Similar conditions are given in Götze and Hipp (1994). Doukhan and Wintenberger (2008) considered the AR(∞) or chain with infinite memory model…”

“…Note that in our setting W (1) = ∞ j=1 w j = 1, while W (1) < 1 is required in Doukhan and Wintenberger (2008). Hence we can allow stronger dependence.…”

“…Hence we can allow stronger dependence. If, as in Doukhan and Wintenberger (2008), W (1) < 1, then a k = O(r k ) for some r ∈ (0, 1). This is analogous to Theorem 2 which ensures the GMC property.…”

Correction to “On weak dependence conditions for Poisson autoregressions” [Statist. Probab. Lett. 82 (2012) 942–948]

“…The above result has been derived by using the general theory of Doukhan and Wintenberger (2008) where the authors consider an infinite memory chain {X t , t ∈ Z} which satisfies the recurrence equation…”

“…To apply the theory of Doukhan and Wintenberger (2008) to the case of model (1), we introduced in the proof of Theorem 2.1 the vector process X t = (Y t , λ t ) and the function F (·) by means of (Y t , λ t ) = (N t (f (X t−1 , . .…”