Infinite divisibility (probability)

Examples of continuous distributions that are infinitely divisible are the normal distribution, the Cauchy distribution, the Lévy distribution, and all other members of the stable distribution family, as well as the Gamma distribution, the chi-square distribution, the Wald distribution, the Log-normal distribution[2] 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 whose only possible outcome is 0 is also (trivially) infinitely divisible.

The uniform distribution and the binomial distribution are not infinitely divisible, nor are any other distributions with bounded support (≈ finite-sized domain), other than the one-point distribution mentioned above.[3] The distribution of the reciprocal of a random variable having a Student's t-distribution is also not infinitely divisible.[4]

Any compound Poisson distribution is infinitely divisible; this follows immediately from the definition.

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

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

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

${\displaystyle \lim _{n\to \infty }\,\max _{1\leq k\leq n}\;P(\left|X_{nk}\right|>\varepsilon )=0{\text{ for every }}\varepsilon >0.}$

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

${\displaystyle \lim _{n\rightarrow \infty }np_{n}=\lambda ,}$

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.

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_n1, X_n2, …, 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.

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

${\displaystyle \{\mu _{t}\}_{t\geq 0}}$ satisfies a number of conditions of continuity and monotonicity. Morover, if a family of infinitely divisible distributions ${\displaystyle \{\mu _{t}\}_{t\geq 0}}$ satisfies these continuity and monotonicity conditions, there exists (uniquely in law) an additive process ${\displaystyle \{\mu _{t}\}_{t\geq 0}}$ with this distribution. [5]

• Cramér's theorem
• Indecomposable distribution
• Compound Poisson distribution

• 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).