Doukhan, Paul; Wintenberger, Olivier Weakly dependent chains with infinite memory. (English) Zbl 1166.60031 Stochastic Processes Appl. 118, No. 11, 1997-2013 (2008). The authors consider the stationary solution of the equation \(X_t = F(X_{t-1},X_{t-2},\ldots;\xi_t)\) a.s. for \(t\in \mathbb{Z}\) where \((\xi_t)_{t\in\mathbb{Z}}\) is a sequence of i.i.d. random variables, \(F\) takes values in a Banach space and satisfies the Lipschitz-type condition. An explicit upper bound for the \(\tau\)-dependence coefficient introduced by J. Dedecker and C. Prieur [J. Theor. Probab. 17, No. 4, 855–885 (2004; Zbl 1067.60008)] of \((X_t)_{t\in\mathbb{Z}}\) is provided under specified assumptions concerning \(F\) (involving certain Orlicz space). The relation between the above mentioned result and the analogous one for mixing coefficients established in [M. Iosifescu, S. Grigorescu, Dependence with Complete Connections and Applications. Cambridge Tracts in Mathematics. 96. (Cambridge), UK: Cambridge University Press. (1990; Zbl 0749.60067)] is discussed. SLLN, CLT and strong invariance principle are obtained using the bounds for \(\tau(r)\) as \(r\to \infty\). Among various examples one can mention non-linear autoregressive models and the Galton – Watson process with immigration. Reviewer: Alexander V. Bulinski (Moskva) Cited in 56 Documents MSC: 60G99 Stochastic processes 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 91B62 Economic growth models 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles Keywords:weak dependence coefficient; SLLN; CLT; strong invariance principle. Citations:Zbl 1067.60008; Zbl 0749.60067 PDFBibTeX XMLCite \textit{P. Doukhan} and \textit{O. 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