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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
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 ...
is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of
independent and identically distributed In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
(i.i.d.) random variables. 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 point ...
of any infinitely divisible distribution is then called an infinitely divisible characteristic function.Lukacs, E. (1970) ''Characteristic Functions'', Griffin , London. p. 107 More rigorously, the probability distribution ''F'' is infinitely divisible if, for every positive integer ''n'', there exist ''n'' i.i.d. random variables ''X''''n''1, ..., ''X''''nn'' whose sum ''S''''n'' = ''X''''n''1 + … + ''X''''nn'' has the same distribution ''F''. The concept of infinite divisibility of probability distributions was introduced in 1929 by
Bruno de Finetti Bruno de Finetti (13 June 1906 – 20 July 1985) was an Italian probabilist statistician and actuary, noted for the "operational subjective" conception of probability. The classic exposition of his distinctive theory is the 1937 "La prévision: ...
. This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for certain models or applications. Infinitely divisible distributions play an important role in probability theory in the context of limit theorems.


Examples

Examples of continuous distributions that are infinitely divisible are 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 i ...
, 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) func ...
, 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 kn ...
, and all other members of the stable distribution family, as well as the
Gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma di ...
, the
chi-square distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squ ...
, the Wald distribution, the
Log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
and the Student's t-distribution. Among the discrete distributions, examples are the Poisson distribution and the negative binomial distribution (and hence the geometric distribution also). The
one-point 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 ...
whose only possible outcome is 0 is also (trivially) infinitely divisible. The
uniform distribution Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence See also * * Homogeneous distribution In mathematics, a homogeneous distribution ...
and the
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no questi ...
are ''not'' infinitely divisible, nor are any other distributions with bounded
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
(≈ finite-sized
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
), other than the
one-point 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 ...
mentioned above. The distribution of the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of a random variable having a Student's t-distribution is also not infinitely divisible. Any
compound Poisson distribution In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable ...
is infinitely divisible; this follows immediately from the definition.


Limit theorem

Infinitely divisible distributions appear in a broad generalization of the
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 themselv ...
: the limit as ''n'' → +∞ of the sum ''S''''n'' = ''X''''n''1 + … + ''X''''nn'' of independent uniformly asymptotically negligible (u.a.n.) random variables within a triangular array : \begin X_ \\ X_ & X_ \\ X_ & X_ & X_ \\ \vdots & \vdots & \vdots & \ddots \end approaches — in the weak sense — an infinitely divisible distribution. The uniformly asymptotically negligible (u.a.n.) condition is given by : \lim_ \, \max_ \; P( \left, X_ \ > \varepsilon ) = 0 \text\varepsilon > 0. Thus, for example, if the uniform asymptotic negligibility (u.a.n.) condition is satisfied via an appropriate scaling of identically distributed random variables with finite variance, the weak convergence is 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 i ...
in the classical version of the central limit theorem. More generally, if the u.a.n. condition is satisfied via a scaling of identically distributed random variables (with not necessarily finite second moment), then the weak convergence is to a stable distribution. On the other hand, for a triangular array of independent (unscaled) Bernoulli random variables where the u.a.n. condition is satisfied through :\lim_ np_n = \lambda, the weak convergence of the sum is to the Poisson distribution with mean ''λ'' as shown by the familiar proof of the law of small numbers.


Lévy process

Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process. A Lévy process is a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
with stationary independent increments, where ''stationary'' means that for ''s'' < ''t'', the
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 ...
of ''L''''t'' − ''L''''s'' depends only on ''t'' − ''s'' and where ''independent increments'' means that that difference ''L''''t'' − ''L''''s'' is independent of the corresponding difference on any interval not overlapping with 's'', ''t'' and similarly for any finite number of mutually non-overlapping intervals. If is a Lévy process then, for any ''t'' ≥ 0, the random variable ''L''''t'' will be infinitely divisible: for any ''n'', we can choose (''X''''n''1, ''X''''n''2, …, ''X''''nn'') = (''L''''t''/''n'' − ''L''0, ''L''2''t''/''n'' − ''L''''t''/''n'', …, ''L''''t'' − ''L''(''n''−1)''t''/''n''). Similarly, ''L''''t'' − ''L''''s'' is infinitely divisible for any ''s'' < ''t''. On the other hand, if ''F'' is an infinitely divisible distribution, we can construct a Lévy process from it. For any interval 's'', ''t''where ''t'' − ''s'' > 0 equals a rational number ''p''/''q'', we can define ''L''''t'' − ''L''''s'' to have the same distribution as ''X''''q''1 + ''X''''q''2 + … + ''X''''qp''. Irrational values of ''t'' − ''s'' > 0 are handled via a continuity argument.


Additive process

An
additive process An additive process, in probability theory, is a cadlag, continuous in probability stochastic process with independent increments. An additive process is the generalization of a Lévy process (a Lévy process is an additive process with identicall ...
\_ (a cadlag, continuous in probability stochastic process with independent increments) has an infinitely divisible distribution for any t\geq 0. Let \_ be its family of infinitely divisible distributions. \_ satisfies a number of conditions of continuity and monotonicity. Morover, if a family of infinitely divisible distributions \_ satisfies these continuity and monotonicity conditions, there exists (uniquely in law) an additive process \_ with this distribution.


See also

* Cramér's theorem *
Indecomposable distribution In probability theory, an indecomposable distribution is a probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables: ''Z'' ≠ ''X'' + ''Y''. ...
*
Compound Poisson distribution In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable ...


Footnotes


References

* Domínguez-Molina, J.A.; Rocha-Arteaga, A. (2007) "On the Infinite Divisibility of some Skewed Symmetric Distributions". ''Statistics and Probability Letters'', 77 (6), 644–648 * Steutel, F. W. (1979), "Infinite Divisibility in Theory and Practice" (with discussion), ''Scandinavian Journal of Statistics.'' 6, 57–64. * Steutel, F. W. and Van Harn, K. (2003), ''Infinite Divisibility of Probability Distributions on the Real Line'' (Marcel Dekker). {{ProbDistributions, Infinite divisibility Theory of probability distributions Types of probability distributions