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

Let ΦnΦn be an i.i.d. sequence of Lipschitz mappings of RdRd. We study the Markov chain {Xnx}n=0∞ on RdRd defined by the recursion Xnx=Φn(Xn−1x), n∈Nn∈N, X0x=x∈Rd. We assume that Φn(x)=Φ(Anx,Bn(x))Φn(x)=Φ(Anx,Bn(x)) for a fixed continuous function Φ:Rd×Rd→RdΦ:Rd×Rd→Rd, commuting with dilations and i.i.d random pairs (An,Bn)(An,Bn), where An∈End(Rd) and BnBn is a continuous mapping of RdRd. Moreover, BnBn is αα-regularly varying and AnAn has a faster decay at infinity than BnBn. We prove that the stationary measure νν of the Markov chain {Xnx} is αα-regularly varying. Using this result we show that, if α<2α<2, the partial sums Snx=∑k=1nXkx, appropriately normalized, converge to an αα-stable random variable. In particular, we obtain new results concerning the random coefficient autoregressive process Xn=AnXn−1+BnXn=AnXn−1+Bn.