Infinitely divisible distribution
An infinitely divisible distribution is a probability distribution that can be represented, for every positive integer (n), as the distribution of a sum of (n) independent and identically distributed random variables. If a probability measure (\mu) on (\mathbb{R}^d) is infinitely divisible, then for each (n\geq 1) there exists a probability measure (\mu_n) satisfying
[ \mu=\underbrace{\mu_n * \mu_n * \cdots * \mu_n}_{n\text{ factors}}, ]
where (*) denotes convolution. The measure (\mu_n) need not belong to the same parametric family as (\mu), and its uniqueness is not part of the definition.
Infinite divisibility is primarily a structural property rather than a statement about the arithmetic divisibility of observed values. It characterizes precisely the one-time marginal distributions of Lévy processes, which are stochastic processes having stationary and independent increments together with an appropriate continuity condition. The concept also provides the measure-theoretic foundation for compound Poisson distributions, convolution semigroups, and many limit laws arising from triangular arrays.
Characteristic-function criterion
Let (X) be an (\mathbb{R}^d)-valued random variable with characteristic function
[ \varphi_X(u)=\mathbb{E}!\left[e^{i\langle u,X\rangle}\right], \qquad u\in\mathbb{R}^d. ]
The distribution of (X) is infinitely divisible if and only if, for every positive integer (n), there exists a characteristic function (\varphi_n) such that
[ \varphi_X(u)=\bigl(\varphi_n(u)\bigr)^n. ]
This condition is stronger than the pointwise existence of complex (n)th roots. A selected root must itself be positive definite, continuous at the origin, and equal to (1) at the origin, as required by Bochner's theorem.
The characteristic function of an infinitely divisible distribution has no zeros. Consequently, it possesses a distinguished continuous logarithm normalized to vanish at the origin. Writing
[ \varphi_X(u)=\exp{\Psi(u)}, ]
the exponent (\Psi) is a continuous negative-definite function. Fractional convolution powers are then defined by
[ \widehat{\mu_t}(u)=\exp{t\Psi(u)}, \qquad t\geq 0, ]
and satisfy the semigroup identity
[ \mu_s * \mu_t=\mu_{s+t}. ]
Thus integer divisibility extends canonically to a continuous convolution semigroup.
Lévy–Khintchine representation
The central classification result is the Lévy–Khintchine formula. Every infinitely divisible distribution on (\mathbb{R}^d) has a characteristic exponent of the form
[ \Psi(u)
i\langle b,u\rangle -\frac{1}{2}\langle u,Au\rangle + \int_{\mathbb{R}^d\setminus{0}} \left( e^{i\langle u,x\rangle} -1 -i\langle u,x\rangle\mathbf{1}_{{\lVert x\rVert\leq 1}} \right)\nu(dx). ]
Here (b\in\mathbb{R}^d) is a drift vector determined relative to the displayed truncation convention. The matrix (A) is symmetric and positive semidefinite, and it describes the Gaussian component. The measure (\nu), called the Lévy measure, satisfies
[ \nu({0})=0, \qquad \int_{\mathbb{R}^d} \min(1,\lVert x\rVert^2),\nu(dx)<\infty. ]
The triplet ((b,A,\nu)) is unique once the truncation function has been fixed. Replacing the indicator-based truncation with another admissible truncation changes the drift parameter but leaves the underlying distribution unchanged.
The representation is conventionally associated with Paul Lévy and Aleksandr Khintchine, whose analyses connected convolution roots with canonical characteristic exponents. Their formulation separates the distribution into deterministic displacement, continuous Gaussian fluctuation, and discontinuous jump behavior. This separation later became the distributional form of the Lévy–Itô decomposition.
Probabilistic interpretation
The Gaussian term corresponds to a possibly degenerate multivariate normal distribution. The integral term records jumps whose sizes are governed by the Lévy measure. Although (\nu) need not have finite total mass, it has finite mass outside every neighborhood of the origin. Large jumps therefore occur with finite intensity over bounded time intervals, while small jumps can occur with infinite activity.
When (\nu(\mathbb{R}^d)<\infty) and the Gaussian term vanishes, the distribution consists of a deterministic shift combined with a compound Poisson component. Its exponent can be written as
[ \Psi(u)
i\langle \gamma,u\rangle + \lambda\int_{\mathbb{R}^d} \left(e^{i\langle u,x\rangle}-1\right)F(dx), ]
where (\lambda=\nu(\mathbb{R}^d)) and (F=\nu/\lambda). The resulting random variable is a sum of a Poisson-distributed number of independent jumps having common distribution (F).
If the Lévy measure has infinite total mass, the jump component generally cannot be expressed as an ordinary finite random sum. It instead arises as a compensated limit of compound Poisson components. The integrability condition in the Lévy–Khintchine formula ensures convergence after the contribution of sufficiently small jumps has been centered.
Relation to Lévy processes
For every infinitely divisible probability measure (\mu), there exists a Lévy process ((X_t)_{t\geq 0}) such that (X_1) has distribution (\mu). Its characteristic functions satisfy
[ \mathbb{E}!\left[e^{i\langle u,X_t\rangle}\right]
\exp{t\Psi(u)}. ]
Conversely, the distribution of (X_t) is infinitely divisible for every (t\geq 0). Indeed, stationary independent increments give
[ X_t
\sum_{k=1}^{n} \left( X_{kt/n}-X_{(k-1)t/n} \right), ]
and the summands are independent with the common distribution of (X_{t/n}).
This correspondence identifies the convolution parameter with elapsed time. It also explains why infinite divisibility concerns decompositions of distributions rather than pathwise subdivision of individual random variables. Different Lévy processes may be realized on different probability spaces, while their one-dimensional laws remain determined by the same convolution semigroup.
Examples and exclusions
Every Gaussian distribution is infinitely divisible. For a normal random variable with mean (m) and variance (\sigma^2), the (n)th convolution root is normal with mean (m/n) and variance (\sigma^2/n).
A Poisson distribution with parameter (\lambda) is infinitely divisible because it is the (n)-fold convolution of Poisson distributions having parameter (\lambda/n). The same structure extends to compound Poisson laws, with the jump intensity divided by (n) in each convolution factor.
The gamma distribution is infinitely divisible when parametrized with positive shape and scale. Dividing the shape parameter by (n) produces an (n)th convolution root while preserving the scale parameter. The associated convolution semigroup is generated by a gamma subordinator.
Every stable distribution is infinitely divisible. Stability imposes the additional requirement that convolution powers agree with affine rescalings of the original law, whereas infinite divisibility alone imposes no such self-similarity.
The uniform distribution on a nondegenerate bounded interval is not infinitely divisible. More generally, a nondegenerate infinitely divisible distribution on (\mathbb{R}^d) cannot have bounded support. Repeated convolution roots would otherwise force a support structure incompatible with the fixed bounded range, except when the measure is concentrated at a single point.
The binomial distribution with a fixed positive number of trials is generally not infinitely divisible. Its probability-generating function has a finite-degree structure that cannot supply convolution roots for every positive integer. This contrasts with the Poisson law, whose generating function is exponential and therefore admits arbitrary positive convolution powers.
Closure and approximation
The class of infinitely divisible distributions is closed under convolution. If (\mu) and (\eta) have characteristic exponents (\Psi_\mu) and (\Psi_\eta), then their convolution has exponent (\Psi_\mu+\Psi_\eta), which again has Lévy–Khintchine form.
The class is also closed under weak convergence, provided the limiting object remains a probability measure. In 1938, You Watanabe established this closure through a canonical analysis of triangular arrays, showing that convergent sequences of infinitely divisible laws retain a Lévy–Khintchine exponent in the limit. The argument identifies convergence of the Gaussian terms, compensated drift terms, and Lévy measures as the three components governing distributional convergence.
The corresponding approximation theory was developed further in William Feller's treatment of limit distributions for infinitesimal triangular arrays. In that setting, each row contains independent random variables whose individual contributions become negligible, while the row sums converge in distribution. Every possible nondegenerate limit of such an infinitesimal array is infinitely divisible, and every infinitely divisible law can be obtained as a limit of this kind.
Compound Poisson approximation gives a direct manifestation of the same principle. Truncating a Lévy measure away from the origin produces a finite measure and therefore a compound Poisson law. After a compensating adjustment to the drift, the resulting distributions converge weakly to the original infinitely divisible distribution as the truncation threshold approaches zero.
Cumulants and moments
Whenever the relevant moments exist, the logarithm of the characteristic function acts as a cumulant-generating object. Under convolution, characteristic exponents add, so finite cumulants add as well. For an (n)th convolution root, each existing cumulant is divided by (n).
Infinite divisibility does not imply the existence of moments. The integrability of large jumps determines the finiteness of many absolute moments, while the behavior of the Lévy measure near the origin controls path variation and small-jump activity. For example, a finite first absolute moment requires an appropriate integrability condition on the portion of the Lévy measure outside the unit ball, together with the corresponding treatment of compensated small jumps.
The tail behavior of an infinitely divisible law is not fixed by infinite divisibility itself. Gaussian components produce rapidly decreasing tails, whereas suitable Lévy measures produce polynomial or heavier decay. The structural constraint lies in the exponent and its measure decomposition rather than in a universal asymptotic form.