Mathematics.

distributions

Infinitely Divisible Distributions

Probability55 minDifficulty8 out of 10

You should know: characteristic functions prob

Overview

A probability distribution is infinitely divisible if, for every positive integer n, it can be expressed as the distribution of the sum of n independent, identically distributed random variables. The class includes the normal, Poisson, gamma, and Cauchy distributions, and all stable distributions. Infinitely divisible distributions are precisely the one-time marginals of Lévy processes — the continuous-time analogues of random walks — making them central to mathematical finance, physics, and the theory of stochastic processes.

Intuition

A distribution is infinitely divisible if you can always split it into equal parts: the sum of n i.i.d. variables (each representing 1/n of the 'total randomness') recovers the original distribution. This is the discrete-time analogue of saying there is a Lévy process — a continuous-time stochastic process with stationary and independent increments — whose distribution at time 1 is exactly this distribution. The Lévy–Khintchine theorem completely characterises infinitely divisible distributions through a Gaussian component (σ²) and a Lévy measure ν describing the jumps.

Formal Definition

Definition

A probability distribution with characteristic function φ is infinitely divisible if for every n ∈ ℕ there exists a characteristic function φₙ such that:

φ(t)=[φn(t)]nfor every n∈N\varphi(t) = [\varphi_n(t)]^n \quad \text{for every } n \in \mathbb{N}
Definition of infinite divisibility
log⁡φ(t)=iγt−σ2t22+∫R∖{0} ⁣(eitx−1−itx1+x2)1+x2x2 ν(dx)\log \varphi(t) = i\gamma t - \frac{\sigma^2 t^2}{2} + \int_{\mathbb{R}\setminus\{0\}} \!\left(e^{itx} - 1 - \frac{itx}{1+x^2}\right)\frac{1+x^2}{x^2}\,\nu(dx)
Lévy–Khintchine representation
Triplet (γ,σ2,ν):γ∈R,  σ2≥0,  ∫Rmin⁡(1,x2) ν(dx)<∞\text{Triplet } (\gamma, \sigma^2, \nu): \quad \gamma \in \mathbb{R},\; \sigma^2 \geq 0,\; \int_{\mathbb{R}} \min(1, x^2)\,\nu(dx) < \infty
Lévy triplet conditions (ν is the Lévy measure)

Worked Examples

  1. 1

    The characteristic function of Poisson(λ) is φ(t) = exp(λ(e^{it} − 1)).

    φ(t)=exp⁡ ⁣(λ(eit−1))\varphi(t) = \exp\!\bigl(\lambda(e^{it}-1)\bigr)
  2. 2

    We need to write φ(t) = [φₙ(t)]ⁿ. Set φₙ(t) = exp((λ/n)(e^{it} − 1)).

    [φn(t)]n=exp⁡ ⁣(n⋅λn(eit−1))=exp⁡ ⁣(λ(eit−1))=φ(t)[\varphi_n(t)]^n = \exp\!\bigl(n \cdot \tfrac{\lambda}{n}(e^{it}-1)\bigr) = \exp\!\bigl(\lambda(e^{it}-1)\bigr) = \varphi(t)
  3. 3

    φₙ is the characteristic function of Poisson(λ/n), which is a valid distribution for every n.

✓ Answer

Poisson(λ) is infinitely divisible: the n-th part is Poisson(λ/n).

Practice Problems

Mediumfree response

Is the Bernoulli(p) distribution infinitely divisible? Justify.

Hardfree response

State the Lévy–Khintchine theorem and identify the role of each component (γ, σ², ν) in the Lévy triplet.

Quiz

Which of the following is NOT infinitely divisible?
The Lévy–Khintchine theorem characterises infinitely divisible distributions via:
Infinitely divisible distributions are precisely the marginal distributions at time 1 of:

Summary

  • A distribution is infinitely divisible if for every n it is the distribution of a sum of n i.i.d. variables.
  • Examples: normal, Poisson, gamma, Cauchy, and all stable distributions. Non-example: Bernoulli.
  • The Lévy–Khintchine theorem: every infinitely divisible distribution has log-characteristic function determined by a triplet (γ, σ², ν).
  • Infinitely divisible distributions are the marginals of Lévy processes — fundamental in finance and physics.

References

  1. WebsiteWikipedia — Infinite divisibility (probability)