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Lévy-Khinchin canonical representation

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A formula for the logarithm of the characteristic function of an infinitely-divisible distribution:

where the integrand is equal to for and the characteristics and are such that is a real number and 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 and in the Lévy–Khinchin canonical representation, and conversely, for any and 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 , , to a distribution (which is necessarily infinitely divisible) with characteristics and it is necessary and sufficient that and that the converge weakly to as .

For references see Lévy canonical representation.


Comments

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

How to Cite This Entry:
Lévy-Khinchin canonical representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%A9vy-Khinchin_canonical_representation&oldid=17693
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article