Positive semidefinite function

A positive semidefinite function is a special complex-valued function that is usually defined on the real numbers or, more generally, on groups . These functions are used, for example, in the formulation of Bochner's theorem , which describes the characteristic functions in stochastics.

definition

One function

${\displaystyle \varphi \colon \mathbb {R} ^{d}\to \mathbb {C} }$

is a positive semi-definite function if for all and all and all , that is true ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle t_{1},t_{2},\dotsc ,t_{n}\in \mathbb {R} ^{d}}$ ${\displaystyle z_{1},z_{2},\ldots ,z_{n}\in \mathbb {C} }$

${\displaystyle \sum _{i=1}^{n}\sum _{k=1}^{n}\varphi (t_{i}-t_{k})\cdot z_{i}\cdot {\overline {z_{k}}}\geq 0}$

is. More generally, it is called a mapping of a group (here written multiplicatively)

${\displaystyle \varphi \colon (G,\cdot )\to \mathbb {C} }$

a positive semidefinite map if the following applies to all and all and all : ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle t_{1},t_{2},\dotsc ,t_{n}\in G}$ ${\displaystyle z_{1},z_{2},\ldots ,z_{n}\in \mathbb {C} }$

${\displaystyle \sum _{i=1}^{n}\sum _{k=1}^{n}\varphi (t_{i}\cdot t_{k}^{-1})\cdot z_{i}\cdot {\overline {z_{k}}}\geq 0}$.

Alternative definition

Alternatively, a positive semidefinite function can be defined as a function in which the matrix for all ${\displaystyle n\in \mathbb {N} }$

${\displaystyle A=(\varphi (t_{i}\cdot t_{k}^{-1}))_{i,k=1,\dots ,n}}$

is a positive semidefinite matrix .

Occur

Positive semidefinite functions occur in stochastics, for example. There it is shown, based on separating families , that the probability measures are uniquely determined by specifying a characteristic function . Thus there is a bijection between the probability measures and the characteristic functions. The set of characteristic functions remains unclear, i.e. for a given function it is not obvious whether it is the characteristic function of a probability measure or not. ${\displaystyle \mathbb {R} ^{d}}$

The set of Bochner describes the characteristic features now fully use the positive semi-definite functions: A continuous function of to is exactly what it is the characteristic function of a probability measure, if it is positive semi-definite and is. ${\displaystyle f}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle f(0)=1}$

Web links

• VS Shul'man: Positive-definite function . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Positive Definite Function . In: MathWorld (English).

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 .