The Fixed Points of the Multivariate Smoothing Transform

Let $N,d>1$ be fixed integers, let $(T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_N)$ 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 := $\mathrm{L}\mathrm{a}\mathrm{w}\mathrm{o}\mathrm{f}$\ (T_1 X_1 + … + T_N X_N + Q)$,wherethe$X_i$areiidwithlaw$\eta$ and independent of $(T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_N,Q)$. Under conditions similar to those for the well-studied case d=1, a complete characterization of all fixed points of $S$ is obtained.