Support, absolute continuity and harmonic moments of fixed points of the multivariate smoothing transform
Jianzhang Mei, Quansheng Liu
TL;DR
This work analyzes fixed points $Z$ of the multivariate smoothing transform with nonnegative random matrices, exploring three key properties: exact support, absolute continuity, and negative harmonic moments. The authors develop a unified framework for both i.i.d. and non-i.i.d. coefficient structures, establishing precise support descriptions via Perron–Frobenius data and cover-sets, proving absolute continuity under a non-degeneracy condition and dimension requirements, and deriving a sharp negative-moment threshold determined by the spectral property $\tilde{\kappa}(-a)$ and $\mathbb{P}(N=1)$. The methods blend probabilistic fixed-point analysis with operator-theoretic and geometric tools (including non-arithmeticity, PF theory, and CD-type support decompositions), yielding results that extend one-dimensional Mandelbrot-cascade insights to the multidimensional setting. Together, these contributions illuminate the density and tail behavior of fixed points and provide practical criteria for both the existence of densities and the finiteness of negative moments, with implications for applications in kinetic models and branching processes in random environments.
Abstract
Consider the multivariate smoothing transform fixed-point equation: $η=$ 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 $η$. For each fixed point $η$, under suitable conditions, we describe its support, establish its absolute continuity, and prove the existence of its harmonic moments.