The Fixed Points of the Multivariate Smoothing Transform

TL;DR

This paper characterizes all fixed points of the multivariate smoothing transform, a probabilistic mapping involving random matrices and vectors, extending known results from the one-dimensional case to higher dimensions.

Contribution

It provides a complete characterization of fixed points for the multivariate smoothing transform under conditions analogous to the one-dimensional case.

Findings

Complete description of fixed points in multivariate setting

Extension of one-dimensional results to higher dimensions

Conditions similar to the scalar case are sufficient

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 $S \eta := \mathrm{Law\ of}\ (T_1 X_1 + ... + T_N X_N + Q)$, 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.