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Fixed Points of the Multivariate Smoothing Transform

Sebastian Mentemeier,
published 2013-09-04

Let N,d > 1 be fixed integers, let (T_1,..., T_N) be random d-by-d matrices with nonnegative entries and Q a random d-vector with nonnegative entries. The multivariate smoothing transform S maps a probability law v on nonnegative vectors to the law of (T_1 X_1 + ... + T_N X_N + Q) where the X_i are iid with law v and independent of (T_1,...,T_N, Q). We give necessary and sufficient conditions for the existence of fixed points which are multivariate regularly varying with index alpha in (0,1) and, as the main result, obtain a full characterization of the set of all fixed points of S. A one-to-one correspondence between fixed points of the homogeneous and inhomogeneous equation is proven, too.

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