2014/09/25 by Kolesko, Konrad, Mentemeier, Sebastian
#60E05 #60G44 #60J80 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1409.7220
Given a sequence (T1, T2, ...) of random d × d matrices with nonnegative entries, suppose there is a random vector X with nonnegative entries, such that ∑i ≥ 1 Ti Xi has the same law as X, where (X1, X2, ...) are i.i.d. copies of X, independent of (T1, T2, ...). Then (the law of) X is called a fixed point of the multivariate smoothing transform. Similar to the well-studied one-dimensional case d=1, a function m is introduced, such that the existence of α∈ (0,1] with m(α)=1 and m'(α) ≤ 0 guarantees the existence of nontrivial fixed points. We prove the uniqueness of fixed points in the critical case m'(α)=0 and describe their tail behavior. This complements recent results for the non-critical multivariate case. Moreover, we introduce the multivariate analogue of the derivative martingale and prove its convergence to a non-trivial limit.