The Fixed Points of the Multivariate Smoothing Transform

Abstract

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. This induces a mapping (the multivariate smoothing transform) on probability laws on the nonnegative cone by $S\eta :=\mathrm{Law}\mathrm{of}(T_1{X}_1+...+T_N{X}_N+Q)$, where the $X_i$ are iid with law $\eta$ and independent of $(T_1,...,T_N,Q)$. Under conditions similar to those for the well-studied case d=1, a complete characterization of all fixed points of $S$ is obtained.

48 pages

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