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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$$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?

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 $AX_1 + BX_2 \stackrel{D}{=} CX$, 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?

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?

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Shanks
Shanks
  • 133
  • 9

Definition and examples of operator-stable distributions

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 $AX_1 + BX_2 \stackrel{D}{=} CX$, 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?