Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems

2010/11/07 by Dariusz Buraczewski, Buraczewski, Dariusz, Ewa Damek +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F05 #60G10 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60G10

paper · pdf · doi:10.48550/arxiv.1011.1685

23 pages, 0 figures. Accepted for publication in Stochastic Processes and their Applications

openalex publication_date 2010/11/07 · arxiv created 2011/10/19 · arxiv updated 2011/10/20 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

Let Φ_n be an i.i.d. sequence of Lipschitz mappings of \R^d. We study the Markov chain \X_n^x\_n=0^∞ on \R^d defined by the recursion X_n^x = Φ_n(X^x_n-1), n∈\N, X₀^x=x∈\R^d. We assume that Φ_n(x)=Φ(A_n x,B_n(x)) for a fixed continuous function Φ:\R^d× \R^d→\R^d, commuting with dilations and i.i.d random pairs (A_n,B_n), where A_n∈ End(\R^d) and B_n is a continuous mapping of \R^d. Moreover, B_n is α-regularly varying and A_n has a faster decay at infinity than B_n. We prove that the stationary measure ν of the Markov chain \X_n^x\ is α-regularly varying. Using this result we show that, if α<2, the partial sums S_n^x=∑_k=1ⁿ X_k^x, appropriately normalized, converge to an α-stable random variable. In particular, we obtain new results concerning the random coefficient autoregressive process X_n = A_n X_n-1+B_n.

Related

• Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps
• Iterated Random Functions and Slowly Varying Tails
• Asymptotically linear iterated function systems on the real line
• Convergence to stable laws for a class of multidimensional stochastic recursions
• Multivariate regular variation of heavy-tailed Markov chains