Stable distribution

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.

Stable

Probability density function Symmetric ${\displaystyle \alpha }$-stable distributions with unit scale factor Skewed centered stable distributions with unit scale factor
Cumulative distribution function CDFs for symmetric ${\displaystyle \alpha }$-stable distributions CDFs for skewed centered stable distributions
Parameters	${\displaystyle \alpha \in (0,2]}$ — stability parameter ${\displaystyle \beta }$ ∈ [−1, 1] — skewness parameter (note that skewness is undefined) c ∈ (0, ∞) — scale parameter μ ∈ (−∞, ∞) — location parameter
Support	x ∈ [μ, +∞) if ${\displaystyle \alpha <1}$ and ${\displaystyle \beta =1}$ x ∈ (-∞, μ] if ${\displaystyle \alpha <1}$ and ${\displaystyle \beta =-1}$ x ∈ R otherwise
PDF	not analytically expressible, except for some parameter values
CDF	not analytically expressible, except for certain parameter values
Mean	μ when ${\displaystyle \alpha >1}$, otherwise undefined
Median	μ when ${\displaystyle \beta =0}$, otherwise not analytically expressible
Mode	μ when ${\displaystyle \beta =0}$, otherwise not analytically expressible
Variance	2c2 when ${\displaystyle \alpha =2}$, otherwise infinite
Skewness	0 when ${\displaystyle \alpha =2}$, otherwise undefined
Ex. kurtosis	0 when ${\displaystyle \alpha =2}$, otherwise undefined
Entropy	not analytically expressible, except for certain parameter values
MGF	${\displaystyle \exp \!\left(t\mu +c^{2}t^{2}\right)}$ when ${\displaystyle \alpha =2}$, otherwise undefined
CF	${\displaystyle \exp \!{\Big [}\;it\mu -|c\,t|^{\alpha }\,(1-i\beta \operatorname {sgn}(t)\Phi )\;{\Big ]},}$ where ${\displaystyle \Phi ={\begin{cases}\tan {\tfrac {\pi \alpha }{2}}&{\text{if }}\alpha \neq 1\\-{\tfrac {2}{\pi }}\log |t|&{\text{if }}\alpha =1\end{cases}}}$

Of the four parameters defining the family, most attention has been focused on the stability parameter, ${\displaystyle \alpha }$ (see panel). Stable distributions have ${\displaystyle 0<\alpha \leq 2}$, with the upper bound corresponding to the normal distribution, and ${\displaystyle \alpha =1}$ to the Cauchy distribution. The distributions have undefined variance for ${\displaystyle \alpha <2}$, and undefined mean for ${\displaystyle \alpha \leq 1}$. The importance of stable probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed (iid) random variables. The normal distribution defines a family of stable distributions. By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that is not normal. Mandelbrot referred to such distributions as "stable Paretian distributions", after Vilfredo Pareto. In particular, he referred to those maximally skewed in the positive direction with ${\displaystyle 1<\alpha <2}$ as "Pareto–Lévy distributions", which he regarded as better descriptions of stock and commodity prices than normal distributions.