Editing Stable distribution (section)

== Properties ==
* Stable distributions are [[infinitely divisible distribution|infinitely divisible]].
* Stable distributions are [[leptokurtotic]] and [[heavy-tailed distribution]]s, with the exception of the [[normal distribution]] (<math>\alpha = 2</math>).
* Stable distributions are closed under [[convolution]].

Stable distributions are closed under convolution for a fixed value of <math>\alpha</math>. Since convolution is equivalent to multiplication of the Fourier-transformed function, it follows that the product of two stable characteristic functions with the same <math>\alpha</math> will yield another such characteristic function. The product of two stable characteristic functions is given by:
<math display="block">\exp\left (it\mu_1+it\mu_2 - |c_1 t|^\alpha - |c_2 t|^\alpha +i\beta_1|c_1 t|^\alpha\sgn(t)\Phi + i\beta_2|c_2 t|^\alpha\sgn(t)\Phi \right )</math>

Since {{math|Φ}} is not a function of the ''μ'', ''c'' or <math>\beta</math> variables it follows that these parameters for the convolved function are given by:
<math display="block">\begin{align}
\mu &=\mu_1+\mu_2 \\
c &= \left (c_1^\alpha+c_2^\alpha \right )^{\frac{1}{\alpha}} \\[6pt]
\beta &= \frac{\beta_1 c_1^\alpha+\beta_2c_2^\alpha}{c_1^\alpha+c_2^\alpha}
\end{align}</math>

In each case, it can be shown that the resulting parameters lie within the required intervals for a stable distribution.