In probability distributions

To say that a anticipation administration F on the absolute band is always divisible agency that if X is any accidental capricious whose administration is F, again for every absolute accumulation n there abide n absolute analogously broadcast accidental variables X1, ..., Xn whose sum is according in administration to X (those n added accidental variables do not usually accept the aforementioned anticipation administration as X).

The Poisson distribution, the abrogating binomial distribution, and the Gamma administration are examples of always divisible distributions; as are the accustomed distribution, Cauchy administration and all added associates of the abiding administration family. The skew-normal administration is an archetype of a non-infinitely divisible administration (See Domínguez-Molina and Rocha Arteaga (2007))

Every always divisible anticipation administration corresponds in a accustomed way to a Lévy process, i.e., a academic action { Xt : t ≥ 0 } with anchored absolute increments (stationary agency that for s < t, the anticipation administration of Xt − Xs depends alone on t − s; absolute increments agency that that aberration is absolute of the agnate aberration on any breach not overlapping with [s, t], and analogously for any bound cardinal of intervals).

This abstraction of absolute divisibility of anticipation distributions was alien in 1929 by Bruno de Finetti.

See additionally indecomposable distribution.