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Revision as of 07:54, 26 March 2012
A formula for the logarithm of the characteristic function of an infinitely-divisible distribution:
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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.
Lévy-Khinchin canonical representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%A9vy-Khinchin_canonical_representation&oldid=22733