Definition and examples of operator-stable distributions

Question

I was trying to understand the basic ideas of the operator-stable distributions. I found the papers by Hudson and Sato. However, unfortunately, I am being unable to understand the mathematical expressions, the authors have used, to define the same. So, to make the definition understandable, I wrote:

If $X_1 \stackrel{D}{=} X_2 \stackrel{D}{=} X$, then $X$ is operator-stable if there exists a real matrix $C$ and a vector $d$ such that $AX_1 + BX_2 \stackrel{D}{=} CX + d$, where $A$, and $B$ are real, square matrices and $X_1$, $X_2$, and $X$ are real vectors.

Have I understood right (only for simple scenarios)? Also, I have not found any examples (like Gaussian, Cauchy etc.) of the operator-stable distributions. Some examples will be really helpful. If there are no such examples available in closed form, then can I find the density numerically at a given point?

Answer 1

$\newcommand{\R}{\mathbb{R}} \newcommand{\D}{\overset{\text{D}}=} \newcommand{\E}{\operatorname{\mathsf E}}$

The fundamental paper in this area is the one by Sharpe. By Theorem 1 in that paper, a full probability measure $\mu$ on $\R^d$ is (operator) stable if and only if for each natural $n$ there are a $d\times d$ real matrix $B_n$ and a vector $b_n\in\R^d$ such that $X_1+\cdots+X_n$ equals $B_nX+b_n$ in distribution, where $X,X_1,X_2,\ldots$ are independent random vectors in $\R^d$ each with distribution $\mu$. (A measure $\mu$ on $\R^d$ is called full if the linear span of its support is $\R^d$.)

If a random vector $X$ in $\R^d$ has a stable distribution, we may say that $X$ itself is stable. Examples of stable random vectors include random vectors with independent stable coordinates. Also, if a a random vector $X$ in $\R^d$ is stable, then any nonsingular affine transformation $BX+b$ of $X$ is also obviously stable (here $B$ is any nonsingular $d\times d$ real matrix and $b\in\R^n$). In particular, any Gaussian distribution on $\R^d$ is stable.

Added: In particular, it follows that the understanding of the notion of operator-stable distributions highlighted in the OP's answer is incorrect, even with the additional assumption that $X_1$ and $X_2$ be independent. Indeed, consider, e.g., the case when $X=\begin{bmatrix}\xi\\ \eta\end{bmatrix}$, where $\xi$ and $\eta$ are independent real-valued random variables with the standard Cauchy and Gaussian distributions, respectively. Then the distribution of $X$ is operator-stable. Let $X_1=\begin{bmatrix}\xi_1\\ \eta_1\end{bmatrix}$ and $X_2=\begin{bmatrix}\xi_2\\ \eta_2\end{bmatrix}$ be independent copies of $X$. Let $A:=\begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $B:=\begin{bmatrix}0&1\\1&0\end{bmatrix}$, and suppose that \begin{equation} Y:=AX_1+BX_2\D CX+d=:Z \tag{1} \end{equation} for some real matrix $C=\begin{bmatrix}c_{11}&c_{12}\\ c_{21}&c_{22}\end{bmatrix}$ and a vector $d$. Then $d=0$, since the distributions of the random vectors $Y$ and $CX$ are symmetric about $0$. We have $Y=\begin{bmatrix}\xi_1+\eta_2\\ \eta_1+\xi_2\end{bmatrix}$, and so, for any $t=\begin{bmatrix}r\\ s\end{bmatrix}\in\R^{2\times1}$, \begin{equation} \E e^{it\cdot Y}=\exp\{-|r|-|s|-r^2/2-s^2/2\}, \end{equation} whereas \begin{equation} \E e^{it\cdot Z}=\exp\{-|rc_{11}+sc_{21}|-(rc_{21}+sc_{22})^2/2\}, \end{equation} so that \begin{equation} |r|+|s|+(r^2+s^2)/2=|rc_{11}+sc_{21}|+(rc_{21}+sc_{22})^2/2 \end{equation} for all real $r,s$. Letting here $r\to\infty$ and $s=0$, we have $c_{21}=\pm1$. Similarly, $c_{22}=\pm1$. On the other hand, letting $r=s\to\infty$, we have $(c_{21}+c_{22})^2/2=1$, which contradicts the conditions $c_{21}=\pm1$ and $c_{22}=\pm1$. This contradiction shows that (1) cannot hold here for any $C,d$.

Answer 2

For a rather detailed description of operator-stable laws, you may want to consult Operator-stable laws by Hudson and Mason.

Examples of course include all stable laws (where all matrices are multiples of the identity matrix). To generate some non-trivial example, observe that operator-stable laws are infinitely divisible laws with a particular structure of the Lévy measure (see Theorems 2 and 3 in the above-mentioned paper). So, for instance, the infinitely divisible law with no Gaussian part, arbitrary drift and the Lévy measure concentrated on a logarithmic spiral: $$ \nu(A) = \int_{-\infty}^\infty e^{-\alpha t} \mathbb{1}_A((e^t \cos(\beta t), e^t \sin(\beta t))) dt $$ is operator-stable (but not stable) when $\alpha \in (0, 2)$ and $\beta \in \mathbb{R}$.