[1] More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n, there exist n i.i.d.
Infinitely divisible distributions play an important role in probability theory in the context of limit theorems.
[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.
random variables within a triangular array approaches — in the weak sense — an infinitely divisible distribution.
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.
On the other hand, for a triangular array of independent (unscaled) Bernoulli random variables where the u.a.n.
condition is satisfied through 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 { Lt : t ≥ 0 } with stationary independent increments, where stationary means that for s < t, the probability distribution of Lt − Ls depends only on t − s and where independent increments means that that difference Lt − Ls 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.
For any interval [s, t] where t − s > 0 equals a rational number p/q, we can define Lt − Ls to have the same distribution as Xq1 + Xq2 + ... + Xqp.
(a cadlag, continuous in probability stochastic process with independent increments) has an infinitely divisible distribution for any
satisfies these continuity and monotonicity conditions, there exists (uniquely in law) an additive process