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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]]:
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A formula for the logarithm $\ln\phi(\lambda)$ 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>
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$$\ln\phi(\lambda)=i\gamma\lambda+\int\limits_{-\infty}^\infty\left(e^{i\lambda x}-1-\frac{i\lambda x}{1+x^2}\right)\frac{1+x^2}{x^2}dG(x),$$
  
where the integrand is equal to <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]].
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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|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" />.
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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 $\gamma$ 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|Lévy canonical representation]].
 
For references see [[Lévy canonical representation|Lévy canonical representation]].

Latest revision as of 17:46, 10 October 2014

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.

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