The defining property of stable distributions

Our group, along with a few students, has been reading statistics handbook and refreshing our understanding of the basic statistics. I was given to cover a chapter about the central limit theorem, which reminded me that I had already given a similar presentation while being PhD student myself. While diving into the topic, I have noticed a couple things, which are usually glanced over in a typical statistics handbook. In the final post of this series, let me put an emphasis on the defining property of any stable distribution.

Namely, if two random variates, \( X \) and \( Y \), are sampled from any specific stable distribution, then their linear combination, e.g.,

\begin{equation} \Sigma = a X + b Y + c, \end{equation}

is also distributed according to the same specific stable distribution. One just needs to adjust the location and the scale parameter values the distribution.

The app below lets you test this property using the Cauchy distribution. The app already has generated some random values for you, but the theoretical (gray) curve doesn't seem to fit the simulated PDF (red curve) very well. Try adjusting parameters of \( \Sigma \) distribution (they "gray" parameters) and observe what happens.

For deeper exploration, adjust the parameter values for the sampling distributions of \( X \) and \( Y \), as well as the summation coefficient values (the "red" parameters). After doing so, press the "Generate" button and try to align the simulated and the theoretical PDFs again.

Observe that the relationship between simulation parameters (the "red" parameters) and theoretical curve parameters (the "gray" parameters) is rather easily predictable. Namely, the location of the linear combination is given by,

\begin{equation} x_0^{(\Sigma)} = a x_0^{(X)} + b x_0^{(Y)} + c . \end{equation}

Here, the superscripts indicate which distribution the parameter relates to. Similarly, for the scale of the linear combination we have,

\begin{equation} \gamma^{(\Sigma)} = a \gamma^{(X)} + b \gamma^{(Y)} . \end{equation}