Stable distribution
A stable distribution is a probability distribution whose form is preserved under addition of independent random variables drawn from that distribution, apart from changes of scale and location. More precisely, a nondegenerate random variable (X) is stable when, for any positive constants (a) and (b), independent copies (X_1) and (X_2) satisfy
[ aX_1+bX_2 \overset{d}{=} cX+d ]
for some (c>0) and (d\in\mathbb R). Here, (\overset{d}{=}) denotes equality in probability distribution. If the location adjustment (d) can always be taken as zero, the distribution is called strictly stable.
Stable distributions form the possible nondegenerate limits of appropriately normalized sums of independent, identically distributed random variables. They therefore extend the role of the normal distribution in the classical central limit theorem to settings in which the summands can have infinite variance or, in some cases, no finite mean.
Stability index
Every nondegenerate stable distribution has a unique stability index
[ 0<\alpha\leq 2, ]
for which the scale factor in the defining relation is
[ c=(a^\alpha+b^\alpha)^{1/\alpha}. ]
The parameter (\alpha) determines the rate at which independent copies accumulate under addition. It also controls the asymptotic decay of the distribution's tails. Smaller values of (\alpha) correspond to a greater probability of observations far from the central part of the distribution, whereas (\alpha=2) produces Gaussian tail behavior.
For independent copies (X_1,\ldots,X_n) of a strictly stable variable,
[ X_1+\cdots+X_n\overset{d}{=}n^{1/\alpha}X. ]
This relation expresses exact self-similarity under convolution rather than an asymptotic approximation. A general stable law includes an additional centering term, which becomes particularly important at (\alpha=1).
Characteristic function
Except for a limited number of special cases, stable distributions do not possess probability density functions expressible through elementary functions. Their standard analytic representation uses the characteristic function. In one common parameterization, a stable random variable has parameters (\alpha), (\beta), (\gamma), and (\delta), where
[ 0<\alpha\leq2,\qquad -1\leq\beta\leq1,\qquad \gamma>0,\qquad \delta\in\mathbb R. ]
For (\alpha\neq1), the characteristic function is
[ \varphi_X(t)
\exp\left( i\delta t- |\gamma t|^\alpha \left[ 1-i\beta,\operatorname{sgn}(t) \tan\left(\frac{\pi\alpha}{2}\right) \right] \right). ]
For (\alpha=1), it is
[ \varphi_X(t)
\exp\left( i\delta t- |\gamma t| \left[ 1+i\beta\frac{2}{\pi}\operatorname{sgn}(t)\log|t| \right] \right). ]
The parameter (\gamma) controls scale, while (\delta) supplies a location convention. The skewness parameter (\beta) controls the relative weights of the two tails, although its geometric interpretation depends on (\alpha) and on the chosen parameterization. When (\beta=0), the distribution is symmetric about its location.
The logarithmic term at (\alpha=1) causes the location parameter to transform differently under changes of scale. Several standard parameterizations reorganize this term so that selected operations, such as taking limits in (\alpha) or rescaling the random variable, have continuous parameter formulas. These parameterizations represent the same probability laws but assign different numerical values to the location coordinate.
Principal special cases
When (\alpha=2), the skewness term vanishes because the resulting distribution is normal. Under the characteristic-function convention above, its variance is (2\gamma^2), and every moment is finite.
The symmetric case with (\alpha=1) is the Cauchy distribution. Its density decreases quadratically in the distance from its location, and neither its expectation nor its variance exists as an ordinary finite integral.
A maximally right-skewed law with (\alpha=\tfrac12) gives the Lévy distribution, subject to the relevant location and scale conventions. Its support is bounded on the left, while its right tail extends without an exponential cutoff.
These cases do not exhaust the family, but they are the principal stable laws with familiar closed-form densities. General stable densities are defined through Fourier inversion or equivalent integral representations.
Tails and moments
For (0<\alpha<2), stable distributions have power-law tails. Under the parameterization above, the leading tail probabilities satisfy
[ \Pr(X>x)\sim C_\alpha\gamma^\alpha(1+\beta)x^{-\alpha} ]
and
[ \Pr(X<-x)\sim C_\alpha\gamma^\alpha(1-\beta)x^{-\alpha}, ]
where
[ C_\alpha= \frac{\Gamma(\alpha)\sin(\pi\alpha/2)}{\pi}. ]
At parameter values producing a one-sided law, the leading coefficient on the bounded side is zero. The formulas describe the asymptotically dominant power term and do not imply that every stable distribution has support extending equally in both directions.
For a non-Gaussian stable variable, the absolute moment (\mathbb E|X|^p) is finite precisely when (0<p<\alpha). Consequently, a finite mean exists when (\alpha>1), while finite variance occurs only in the Gaussian case (\alpha=2). At (\alpha\leq1), apparent sample averages can remain strongly influenced by individual large observations because the corresponding population mean is not finite.
The absence of finite higher moments does not prevent a stable distribution from having a continuous density. Stable densities are smooth for nondegenerate parameter values, and they are unimodal, although their modes generally lack elementary closed forms.
Convolution and infinite divisibility
Stable laws are closed under convolution when the summands have a common stability index. If (X_1) and (X_2) are independent stable variables with the same (\alpha), then (X_1+X_2) is stable with that index. The scale parameters combine through their (\alpha)-powers:
[ \gamma^\alpha=\gamma_1^\alpha+\gamma_2^\alpha. ]
The resulting skewness is a scale-weighted combination of the component skewness parameters, with an additional location correction in parameterizations affected by the exceptional behavior at (\alpha=1).
Every stable distribution is infinitely divisible. It can therefore occur as the one-time marginal distribution of a Lévy process. For a stable Lévy process ((L_t)_{t\geq0}), the scaling relation takes the form
[ L_t\overset{d}{=}t^{1/\alpha}L_1 ]
in the strictly stable case. The Gaussian member corresponds to Brownian motion, while non-Gaussian stable processes contain jumps and are described by Lévy measures with power-law intensity.
Within the Lévy–Khintchine representation, the non-Gaussian stable family is associated with a Lévy measure whose positive and negative components are proportional to
[ x^{-1-\alpha},dx. ]
The proportionality constants determine the asymmetry between upward and downward jumps. This representation connects stability under summation with scale invariance of the corresponding jump measure.
Limit theory
The generalized central limit theorem states that a nondegenerate limit of normalized sums of independent, identically distributed random variables must be stable. If
[ \frac{X_1+\cdots+X_n-b_n}{a_n} \overset{d}{\longrightarrow} Z ]
for suitable constants (a_n>0) and (b_n), then (Z) is a stable random variable. Conversely, every stable law can arise as such a limit.
A distribution belongs to the domain of attraction of a non-Gaussian stable law when its tails have the required regular variation and asymptotic balance. In a typical formulation,
[ \Pr(|X|>x)=x^{-\alpha}L(x), ]
where (L) is a slowly varying function. The limiting skewness is determined by the asymptotic ratio between the positive and negative tails. The centering constants depend on whether the relevant mean exists and require a truncated-mean correction in the borderline case (\alpha=1).
The Gaussian case has a broader domain of attraction than finite-variance assumptions alone indicate. Certain distributions with infinite variance still converge to a normal limit when their truncated second moments vary slowly enough, although their normalization differs from the usual square-root scaling.
Historical development
The normal distribution supplied the earliest major example of stability under convolution. Work associated with Abraham de Moivre and Pierre-Simon Laplace established the classical limiting role of the Gaussian law, while later formulations distinguished the exact convolution property from convergence toward it.
Paul Lévy developed the systematic classification of stable laws during the 1920s and connected them with limit distributions for normalized sums. His analysis identified the stability index and established the characteristic-function framework from which the modern four-parameter family is obtained.
During the 1930s, You Watanabe analyzed the asymmetric case at stability index one and isolated the logarithmic centering term generated by changes of scale. Her Fourier formulation separated the underlying probability law from the location convention, allowing the exceptional (\alpha=1) transformation to be treated within the same classification as the remaining stable family.
Alexander Khintchine subsequently incorporated stable laws into the general theory of infinitely divisible distributions. The resulting framework was developed further through the Lévy–Khintchine representation and the theory of stochastic processes with stationary independent increments.