Lévy Khinchin formula

The Lévy-Khinchin formula is a mathematical theorem from probability theory . It characterizes the infinitely divisible probability distributions on the real numbers via a canonical representation of their logarithmized characteristic function , which consists of three parts.

The Lévy-Khinchin formula is based on a work by Paul Lévy from 1934, which generalizes a formula by Andrei Nikolajewitsch Kolmogorow from 1932. In 1937, Alexander Yakovlevich Chintschin published the Lévy-Khinchin formula.

The Lévy-Khinchin formula is important, for example, for the theory of the Lévy processes , since a corresponding decomposition for the Lévy processes can be derived from the representation of the logarithmic characteristic function as three parts.

statement

Be a probability measure on with characteristic function . Define ${\displaystyle P}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \varphi (t)}$

${\displaystyle \Psi (t):=\ln \varphi (t)}$.

Then:

${\displaystyle P}$is infinitely divisible if and only if there is a real number and a positive number as well as a σ-finite measure for which and ${\displaystyle b}$ ${\displaystyle \sigma ^{2}}$ ${\displaystyle \mu }$ ${\displaystyle \mu (\{0\})=0}$

${\displaystyle \int _{\mathbb {R} }\min(x^{2},1)\mu (\mathrm {d} x)<\infty }$

holds, so the representation ${\displaystyle \Psi }$

${\displaystyle \Psi (t)=ibt-{\frac {\sigma ^{2}}{2}}t^{2}+\int _{\mathbb {R} }\left(\exp(itx)-1-itx\mathbf {1} _{\{|x|<1\}}\right)\mu (\mathrm {d} x)}$

owns.

Here the indicator function denotes the quantity . ${\displaystyle \mathbf {1} _{A}}$ ${\displaystyle A}$

The measure is called the canonical measure or Lévy measure of , the number as the centering constant and the Gaussian coefficient. Together they are called a canonical triple . ${\displaystyle \mu }$ ${\displaystyle P}$ ${\displaystyle b}$ ${\displaystyle \sigma ^{2}}$ ${\displaystyle (\sigma ^{2},b,\mu )}$

Every infinitely divisible probability distribution has a clearly defined canonical triple. Conversely, given a canonical triple, a unique, infinitely divisible probability distribution can be constructed.

Web links

• Lévy-Khinchin canonical representation . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .

Individual evidence

1. ^ BA Rogozin: Lévy canonical representation . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
2. ↑ Klenke: Probability Theory. 2013, p. 345.