In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, a
distribution Distribution may refer to:
Mathematics
* Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a vari ...
is said to be stable if a
linear combination of two
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s with this distribution has the same distribution,
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
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.
[B. Mandelbrot, The Pareto–Lévy Law and the Distribution of Income, International Economic Review 1960 https://www.jstor.org/stable/2525289]
Of the four parameters defining the family, most attention has been focused on the stability parameter,
(see panel). Stable distributions have
, with the upper bound corresponding to the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, and
to the
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
. The distributions have undefined
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
for
, and undefined
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
for
}. 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
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
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
Vilfredo Federico Damaso Pareto ( , , , ; born Wilfried Fritz Pareto; 15 July 1848 – 19 August 1923) was an Italians, Italian polymath (civil engineer, sociologist, economist, political scientist, and philosopher). He made several important ...
. In particular, he referred to those maximally skewed in the positive direction with
as "Pareto–Lévy distributions",
which he regarded as better descriptions of stock and commodity prices than normal distributions.
[Mandelbrot, B., New methods in statistical economics ]The Journal of Political Economy
The ''Journal of Political Economy'' is a monthly peer-reviewed academic journal published by the University of Chicago Press. Established by James Laurence Laughlin in 1892, it covers both theoretical and empirical economics. In the past, th ...
, 71 #5, 421–440 (1963).
Definition
A non-
degenerate distribution
In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter d ...
is a stable distribution if it satisfies the following property:
Since the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, the
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
, and the
Lévy distribution
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
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 distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
s parametrized by location and scale parameters ''μ'' and ''c'', respectively, and two shape parameters
and
, roughly corresponding to measures of asymmetry and concentration, respectively (see the figures).
The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of any probability distribution is just the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of its probability density function
. The density function is therefore the inverse Fourier transform of the characteristic function:
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
where is just the
sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or ...
of and
''μ'' ∈ R is a shift parameter,