Lévy-Khinchin canonical representation

A formula for the logarithm $\ln\phi(\lambda)$ of the characteristic function of an infinitely-divisible distribution:

$$\ln\phi(\lambda)=i\gamma\lambda+\int\limits_{-\infty}^\infty\left(e^{i\lambda x}-1-\frac{i\lambda x}{1+x^2}\right)\frac{1+x^2}{x^2}dG(x),$$

where the integrand is equal to $-\lambda^2/2$ for $x=0$ and the characteristics $\gamma$ and $G$ are such that $\gamma$ is a real number and $G$ is a non-decreasing left-continuous function of bounded variation.

The Lévy–Khinchin canonical representation was proposed by A.Ya. Khinchin (1937) and is equivalent to a formula proposed a little earlier by P. Lévy (1934) and called the Lévy canonical representation. To each infinitely-divisible distribution corresponds a unique set of characteristics $\gamma$ and $G$ in the Lévy–Khinchin canonical representation, and conversely, for any $\gamma$ and $G$ as above, the Lévy–Khinchin canonical representation determines the logarithm of the characteristic function of an infinitely-divisible distribution. For the weak convergence of the sequence of infinitely-divisible distributions determined by characteristics $\gamma_n$, $G_n$, $n=1,2,\dots,$ to a distribution (which is necessarily infinitely divisible) with characteristics $\gamma$ and $G$ it is necessary and sufficient that $\lim\gamma_n=\gamma$ and that the $G_n$ converge weakly to $G$ as $n\to\infty$.

For references see Lévy canonical representation.

Comments

For the notion of weak convergence see Distributions, convergence of.