Infinitely divisible laws are precisely the possible weak limits of sums from infinitesimal independent arrays. Stable laws are the self-similar fixed points of such summation. Together they extend Gaussian limit theory to jump limits and heavy tails, and they provide the one-time marginals of Levy processes.
Convolution roots and examples
Definition. A probability law \(\mu\) on \(\mathbb R^d\) is infinitely divisible if, for every \(n\ge1\), there is a law \(\mu_n\) such that \[ \mu=\mu_n^{*n}. \] Equivalently, a random vector \(X\sim\mu\) can, for every \(n\), be represented in distribution as a sum of \(n\) i.i.d. random vectors.
Example. Gaussian laws are infinitely divisible because \[ N_d(m,\Sigma) = \mathcal L\!\left(\sum_{k=1}^nY_k\right), \qquad Y_k\stackrel{\mathrm{i.i.d.}}{\sim} N_d(m/n,\Sigma/n). \] Poisson laws are infinitely divisible by additivity of independent Poisson variables.
Proposition (compound Poisson law). Let \(N\sim\operatorname{Poisson}(\lambda)\), let \(Y_j\) be i.i.d. with law \(Q\), independent of \(N\), and put \(X=\sum_{j=1}^NY_j\). Then \[ \mathbb Ee^{it^\top X} =\exp\left\{\lambda\int_{\mathbb R^d} (e^{it^\top x}-1)Q(\mathrm{d} x)\right\}, \] and the law of \(X\) is infinitely divisible.
Proof. Conditional on \(N\), \[ \mathbb E[e^{it^\top X}\mid N] =\varphi_Q(t)^N. \] Taking expectation and using the Poisson generating function gives \[ \mathbb E\varphi_Q(t)^N =\exp\{\lambda(\varphi_Q(t)-1)\}. \] Dividing \(\lambda\) by \(n\) produces an \(n\)th convolution root.
The Levy-Khintchine representation
Definition. A measure \(\nu\) on \(\mathbb R^d\setminus\{0\}\) is a Levy measure if \[ \int_{\mathbb R^d}(1\wedge\|x\|^2)\nu(\mathrm{d} x)<\infty. \] It may have infinite mass near zero, but only enough small-jump mass for the truncated second moment to remain finite.
Theorem (Levy-Khintchine). A law on \(\mathbb R^d\) is infinitely divisible if and only if its characteristic function is \[ \varphi(t)=e^{\psi(t)}, \] where \[ \psi(t) =it^\top b-\frac12t^\top At +\int_{\mathbb R^d} \left(e^{it^\top x}-1-it^\top x\,1_{\{\|x\|\le1\}}\right) \nu(\mathrm{d} x). \tag{LK} \] Here \(b\in\mathbb R^d\), \(A\) is symmetric nonnegative definite, and \(\nu\) is a Levy measure. For the fixed truncation \(x1_{\{\|x\|\le1\}}\), the triplet \((b,A,\nu)\) is unique.
Proof outline. The drift and Gaussian terms are characteristic exponents of infinitely divisible laws. If \(\nu\) is finite away from zero, the integral is the exponent of a compound Poisson law. Truncate an arbitrary Levy measure to \(\{\varepsilon<\|x\|\le R\}\); the corresponding compensated compound Poisson laws are infinitely divisible, and the integrability of \(1\wedge\|x\|^2\) makes their exponents converge locally uniformly to (LK). Levy continuity gives the forward construction. Conversely, take convolution roots \(\mu_n\) and view \(\mu=\mu_n^{*n}\) as a row sum of an infinitesimal array. Compactness of truncated second-moment measures extracts a Gaussian covariance at zero and a vague limit \(\nu\) away from zero. The truncated first moments determine \(b\). Taylor expansion of the row characteristic functions then yields (LK). Uniqueness follows by recovering \(\nu\) from the exponent away from zero, \(A\) from the residual quadratic form, and \(b\) from the linear term.
The three terms have distinct probabilistic meanings: \(b\) is deterministic drift, \(A\) is continuous Gaussian fluctuation, and \(\nu\) records jump intensity by size and direction. Changing the truncation convention changes \(b\), but not \(A\), \(\nu\), or the law.
Limits of infinitesimal arrays
Theorem. Every weak limit of row sums of an infinitesimal row-wise independent triangular array is infinitely divisible. Conversely, every infinitely divisible law arises as such a limit.
Proof outline. Partition each row into \(m\) consecutive blocks whose maximal contribution vanishes. Along a subsequence, the block sums have asymptotically identical convolution-scale contributions, so the limiting characteristic function admits an \(m\)th root for every \(m\). This proves infinite divisibility. For the converse, if \(\mu=\mu_n^{*n}\), use a row of \(n\) i.i.d. variables with law \(\mu_n\); the roots become infinitesimal after an appropriate centering convention, and every row sum has law \(\mu\).
Stable laws
Definition. A nondegenerate law is stable if, for independent copies \(X_1,X_2\) and every \(a,b>0\), there exist \(c>0\) and \(d\in\mathbb R\) such that \[ aX_1+bX_2\stackrel d=cX+d. \] It is strictly stable if \(d\) can always be chosen as zero.
Theorem (one-dimensional parameterization). A nondegenerate stable law has an index \(\alpha\in(0,2]\), scale \(\sigma>0\), skewness \(\beta\in[-1,1]\), and location \(\mu\in\mathbb R\). In a standard parameterization, for \(\alpha\ne1\), \[ \varphi(t) = \exp\!\left\{ i\mu t-\sigma^\alpha|t|^\alpha \left(1-i\beta\,\operatorname{sgn}(t) \tan\frac{\pi\alpha}{2}\right) \right\}, \] while for \(\alpha=1\), \[ \varphi(t) = \exp\!\left\{ i\mu t-\sigma|t| \left(1+i\beta\frac2\pi\operatorname{sgn}(t)\log|t|\right) \right\}. \] For i.i.d. copies, \(X_1+\cdots+X_n\) has the same stable shape after scaling by \(n^{1/\alpha}\), with a location adjustment when necessary.
Proof outline. Stability turns convolution into the functional equation \[ \psi(at)+\psi(bt)=\psi(ct)+idt \] for the log characteristic exponent. Continuity and associativity force \(c^\alpha=a^\alpha+b^\alpha\) for a unique \(\alpha\in(0,2]\). Homogeneity in the Levy-Khintchine representation then forces the Levy measure to have radial density proportional to \(r^{-1-\alpha}\), with separate masses in the positive and negative directions. Evaluating the compensated integral gives the displayed formulas. At \(\alpha=2\), the jump term collapses to a Gaussian quadratic exponent.
Examples. The Gaussian law is stable with \(\alpha=2\). The standard Cauchy law has \(\alpha=1\), \(\beta=0\), and characteristic function \(e^{-|t|}\). A one-sided Levy distribution has \(\alpha=1/2\) and maximal positive skewness. Except in a few cases, stable densities have no elementary closed form even though their characteristic functions are explicit.
Domains of attraction and regular variation
Theorem (generalized central limit theorem). If there exist constants \(a_n>0\), \(b_n\in\mathbb R\) such that \[ \frac{X_1+\cdots+X_n-b_n}{a_n}\Rightarrow Z \] for i.i.d. \(X_i\) and a nondegenerate \(Z\), then \(Z\) is stable. Conversely, every stable law occurs as such a limit.
Proof. Split a sum of \(mn\) observations into \(m\) independent blocks of size \(n\). Any weak limit of the normalized full sum must agree, after the induced scaling and centering, with a sum of \(m\) independent copies of the block limit. This is the stability identity. Conversely, sums of i.i.d. stable variables reproduce the same law under their defining normalization.
Proposition (heavy-tail domain of attraction). Suppose, for some \(\alpha\in(0,2)\) and slowly varying \(L\), \[ P(X>x)\sim c_+x^{-\alpha}L(x),\qquad P(X<-x)\sim c_-x^{-\alpha}L(x), \] with \(c_++c_->0\). Choose \(a_n\) so that \(nP(|X|>a_n)\to1\), and use the appropriate truncated-mean centering \(b_n\). Then \[ \frac{S_n-b_n}{a_n}\Rightarrow Z_\alpha \] for an \(\alpha\)-stable law whose skewness is determined by \(c_+\) and \(c_-\).
Remark. For a non-Gaussian \(\alpha\)-stable variable, \[ \mathbb E|X|^p<\infty\quad\Longleftrightarrow\quad0<p<\alpha. \] Thus variance is infinite whenever \(\alpha<2\), and the mean is absent when \(\alpha\le1\) except for special principal-value symmetries. The normalization \(n^{1/\alpha}\), modified by a slowly varying factor, replaces \(\sqrt n\).
Levy processes
Theorem. A stochastic process \((L_t)_{t\ge0}\) with stationary independent increments, \(L_0=0\), and stochastic continuity satisfies \[ \mathbb Ee^{iu^\top L_t}=e^{t\psi(u)} \] for a Levy-Khintchine exponent \(\psi\). Conversely, every Levy triplet defines such a process with a cadlag modification.
Brownian motion corresponds to a purely Gaussian triplet; a Poisson process to a finite jump measure; and a stable Levy process to a homogeneous jump measure. This decomposition is the jump-process analogue of separating drift, quadratic variation, and compensated noise in stochastic calculus.