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. Of the four parameters defining the family, most attention has been focused on the stability parameter, α {\displaystyle \alpha } (see panel). Stable distributions have 0 < α ≤ 2 {\displaystyle 0<\alpha \leq 2} , with the upper bound corresponding to the normal distribution, and approaches the Dirac delta function in the limit as α → 0 {\displaystyle \alpha \rightarrow 0} . The distributions have undefined variance for α < 2 {\displaystyle \alpha <2} , and undefined mean for α ≤ 1 {\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 1 < α < 2 {\displaystyle 1<\alpha <2} as "Pareto–Lévy distributions", which he regarded as better descriptions of stock and commodity prices than normal distributions.
Definition A non-degenerate distribution is a stable distribution if it satisfies the following property:
Since the normal distribution, the Cauchy distribution, and the Lévy distribution all have the above property, it follows that they are special cases of stable distributions. Such distributions form a four-parameter family of continuous probability distributions parametrized by location and scale parameters μ and c, respectively, and two shape parameters β {\displaystyle \beta } and α {\displaystyle \alpha } , roughly corresponding to measures of asymmetry and concentration, respectively (see the figures). The characteristic function φ {\displaystyle \varphi } of a probability distribution with density function f {\displaystyle f} is the Fourier transform of f . {\displaystyle f.} The density function is then the inverse Fourier transform of the characteristic function:
φ ( t ) = ∫ − ∞ ∞ f ( x ) e i x t d x . {\displaystyle \varphi (t)=\int _{-\infty }^{\infty }f(x)e^{ixt}\,dx.}
Although the probability density function for a general stable distribution cannot be written analytically, the general characteristic function can be expressed analytically. A random variable X is called stable if its characteristic function can be written as
φ ( t ; α , β , c , μ ) = exp ( i t μ − | c t | α ( 1 − i β sgn ( t ) Φ ) ) {\displaystyle \varphi (t;\alpha ,\beta ,c,\mu )=\exp \left(it\mu -|ct|^{\alpha }\left(1-i\beta \operatorname {sgn}(t)\Phi \right)\right)}
where sgn(t) is just the sign of t and
Φ = { tan ( π α 2 ) α ≠ 1 − 2 π log | t | α = 1 {\displaystyle \Phi ={\begin{cases}\tan \left({\frac {\pi \alpha }{2}}\right)&\alpha \neq 1\\-{\frac {2}{\pi }}\log |t|&\alpha =1\end{cases}}}
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