View source for Lévy-Khinchin canonical representation

A formula for the logarithm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l0583001.png" /> of the [[Characteristic function|characteristic function]] of an [[Infinitely-divisible distribution|infinitely-divisible distribution]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l0583002.png" /></td> </tr></table>

where the integrand is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l0583003.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l0583004.png" /> and the characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l0583005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l0583006.png" /> are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l0583007.png" /> is a real number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l0583008.png" /> is a non-decreasing left-continuous [[Function of bounded variation|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|Lévy canonical representation]]. To each infinitely-divisible distribution corresponds a unique set of characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l0583009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830010.png" /> in the Lévy–Khinchin canonical representation, and conversely, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830012.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830015.png" /> to a distribution (which is necessarily infinitely divisible) with characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830017.png" /> it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830018.png" /> and that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830019.png" /> converge weakly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830020.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058300/l05830021.png" />.

For references see [[Lévy canonical representation|Lévy canonical representation]].



====Comments====
For the notion of weak convergence see [[Distributions, convergence of|Distributions, convergence of]].