Infinite divisibility (probability)

In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function.[1]

More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n, there exist n independent identically distributed random variables X_n1, ..., X_nn whose sum S_n = X_n1 + … + X_nn has the distribution F.

The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. 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.[1]

Examples

The Poisson distribution, the negative binomial distribution (and therefore also the geometric distribution), the Gamma distribution and the degenerate distribution are examples of infinitely divisible distributions; as are the normal distribution, Cauchy distribution and all other members of the stable distribution family. The uniform distribution and the binomial distribution are not infinitely divisible, nor are any other (non-trivial) distributions with bounded (finite) support.[2] The Student's t-distribution is infinitely divisible, while the distribution of the reciprocal of a random variable having a Student's t-distribution, is not.[3]

All the compound Poisson distributions are infinitely divisible.

Limit theorem

Infinitely divisible distributions appear in a broad generalization of the central limit theorem: the limit as n → +∞ of the sum S_n = X_n1 + … + X_nn of independent uniformly asymptotically negligible (u.a.n.) random variables within a triangular array

{\begin{array}{cccc}X_{11}\\X_{21}&X_{22}\\X_{31}&X_{32}&X_{33}\\\vdots &\vdots &\vdots &\ddots \end{array}}

approaches — in the weak sense — an infinitely divisible distribution. The uniformly asymptotically negligible (u.a.n.) condition is given by

\lim _{n\to \infty }\,\max _{1\leq k\leq n}\;P(\left|X_{nk}\right|>\varepsilon )=0{\text{ for every }}\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 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 _{n\rightarrow \infty }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 { L_t : t ≥ 0 } with stationary independent increments, where stationary means that for s < t, the probability distribution 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 { L_t : t ≥ 0 } 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_n0, X_n1, …, X_nn) = (L_t/n − L₀, L_2t/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 { L_t : t ≥ 0 } 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_q1 + X_q2 + … + X_qp. Irrational values of t − s > 0 are handled via a continuity argument.

Additive process

An additive process $\{X_{t}\}_{t\geq 0}$ (a cadlag, continuous in probability stochastic process with independent increments) has infinitely divisible distribution for any $t\geq 0$ . Let $\{\mu _{t}\}_{t\geq 0}$ be its family of infinitely divisible distribution.

$\{\mu _{t}\}_{t\geq 0}$ satisfies a number of conditions of continuity and monotonicity. Morover, if a family of infinitely divisible distribution $\{\mu _{t}\}_{t\geq 0}$ satisfies the same continuity and monotonicity conditions it exists (uniquely in law) an Additive process with this distribution $\{\mu _{t}\}_{t\geq 0}$ . [4]

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gollark: I return to the store, in case it restocked.

Footnotes

Lukacs, E. (1970) Characteristic Functions, Griffin , London. p. 107
Sato, Ken-iti (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. p. 31, 148. ISBN 978-0-521-55302-5.
Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, 2nd Edition. Wiley, ISBN 0-471-58494-0 (Chapter 28, page 368)
Sato, Ken-Ito (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press. pp. 31–68. ISBN 9780521553025.

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 doi:10.1016/j.spl.2006.09.014
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).

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[Lukacs-1] Lukacs, E. (1970) Characteristic Functions, Griffin , London. p. 107

[2] Sato, Ken-iti (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. p. 31, 148. ISBN 978-0-521-55302-5.

[3] Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, 2nd Edition. Wiley, ISBN 0-471-58494-0 (Chapter 28, page 368)

[4] Sato, Ken-Ito (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press. pp. 31–68. ISBN 9780521553025.

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped