Moments for multi-dimensional Mandelbrot's cascades

Abstract

We consider the distributional equation $\textbf{Z}\stackrel{d}{=}\sum_{k=1}^N\textbf{A}_k\textbf{Z}(k) $, where $N$ is a random variable taking value in $\mathbb N_0=\{0,1,\cdots\}$, $\textbf{A}_1,\textbf{A}_2,\cdots$ are $p\times p$ non-negative random matrix, and $\textbf{Z},\textbf{Z}(1),\textbf{Z}(2),\cdots$ are $i.i.d$ random vectors in in $\mathbb{R}_+^p$ with $\mathbb{R}_+=[0,\infty)$, which are independent of $(N,\textbf{A}_1,\textbf{A}_2,\cdots)$. Let $\{\mathbf Y_n\}$ be the multi-dimensional Mandelbrot's martingale defined as sums of products of random matrixes indexed by nodes of a Galton-Watson tree plus an appropriate vector. Its limit $\mathbf Y$ is a solution of the equation above. For $\alpha>1$, we show respectively a sufficient condition and a necessary condition for $\mathbb E\|\mathbf Y\|^\alpha\in(0,\infty)$. Then for a non-degenerate solution $\mathbf Z$ of the equation above, we show the decay rates of $\mathbb E e^{-\mathbf t\cdot \mathbf Z}$ as $\|\mathbf t\|\rightarrow\infty$ and those of the tail probability $\mathbb P(\mathbf y\cdot \mathbf Z\leq x)$ as $x\rightarrow 0$ for given $\mathbf y=(y^1,\cdots,y^p)\in \mathbb R_{+}^p$, and the existence of the harmonic moments of $\mathbf y\cdot \mathbf Z$. As application, these above results about the moments (of positive and negative orders) of $\mathbf Y$ are applied to a special multitype branching random walk. Moreover, for the case where all the vectors and matrixes of the equation above are complex, a sufficient condition for the $L^\alpha$ convergence and the $\alpha$th-moment of the Mandelbrot's martingale $\{\mathbf Y_n\}$ is also established.