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

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 n − 1 x ) , n ∈ N , X 0 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 = 1 n 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 .