Tail behaviour of stationary solutions of random difference equations: the case of regular matrices

Given a sequence (M-n, Q(n))n(>= 1) of i.i.d. random variables with generic copy (M, Q) such that M is a regular d X d matrix and Q takes values in R-d, we consider the random difference equation R-n = MnRn-1 + Q(n), n >= 1. Under suitable assumptions stated below, this equation has a unique stationary solution R such that for some kappa > 0 and some finite positive and continuous function K on Sd-1 := {x is an element of R-d : vertical bar x vertical bar = 1}, lim(t ->infinity)t(kappa) P(xR > t) = K(x) for all x is an element of Sd-1 holds true. A rather long proof of this result, originally stated by Kesten [Acta Math. 131 (1973), pp. 207-248] at the end of his famous article, was given by Le Page [Seminaires de probabilites Rennes 1983, University of Rennes I, Rennes, 1983, p. 116]. The purpose of this article is to show how regeneration methods can be used to provide a much shorter argument (particularly for the positivity of K). It is based on a multidimensional extension of Goldie's implicit renewal theory developed in Goldie [Ann. Appl. Probab. 1 (1991), pp. 126-166].