Support, absolute continuity and harmonic moments of fixed points of the multivariate smoothing transform

TL;DR

This paper investigates the properties of fixed points of a multivariate smoothing transform, focusing on their support, absolute continuity, and harmonic moments under certain conditions.

Contribution

It provides a comprehensive analysis of fixed points of the multivariate smoothing transform, including support characterization, absolute continuity, and harmonic moments existence.

Findings

Support of fixed points characterized

Absolute continuity established

Harmonic moments proven to exist

Abstract

Consider the multivariate smoothing transform fixed-point equation: $\eta =$ law of $ \sum_{i=1}^N A_i Z_i$, where $N \geq 0$ is a random integer, $(A_i)_{i \geq 1}$ are $d \times d$ random nonnegative matrices, $(Z_i)_{i \geq 1}$ is a sequence of $\mathbb{R}_+^d$-valued random variables independent of $(N, A_1, A_2, \cdots)$, and all $Z_i$ have the same law $\eta$. For each fixed point $\eta$, under suitable conditions, we describe its support, establish its absolute continuity, and prove the existence of its harmonic moments.