Precise Tail Asymptotics for Attracting Fixed Points of Multivariate Smoothing Transformations

Given d ≥ 1, let (Ai)i≥1 be a sequence of random d × d real matrices and Q be a random vector in Rd. We consider fixed points of multivariate smoothing transforms, i.e. random variables X ∈ Rd satisfying X has the same law as ∑ i≥1 AiXi + Q, where (Xi)i≥1 are i.i.d. copies of X and independent of (Q, (Ai)i≥1). The existence of fixed points that can attract point masses can be shown by means of contraction arguments. Let X be such a fixed point. Assuming that the action of the matrices is expanding as well as contracting with positive probability, it was shown in a number of papers that there is β > 0 with limt→∞ tβ P (〈u, X〉 > t) = K · f (u), where u denotes an arbitrary element of the unit sphere and f a positive function and K ≥ 0. However in many cases it was not established that K is indeed positive. In this paper, under quite general assumptions, we prove that lim inf t→∞ tβ P (〈u, X〉 > t) > 0, completing, in particular, the results of Mirek (2013) and Buraczewski et al. (2013).