Correspondence record (stochastics)

The correspondence theorem is a mathematical proposition from stochastics . It provides a close connection between probability distributions on the real numbers and the distribution functions . This connection allows distribution functions to be examined instead of probability distributions. These are more easily accessible as real functions than the probability distributions, which are set functions on a complex set system, the Borel σ-algebra .

The correspondence set is a consequence of the measure uniqueness theorem .

preparation

A probability distribution is given on the real numbers, i.e. the measurement space . In this article a distinction is made between the distribution function of a probability distribution, which is called ${\displaystyle P}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

${\displaystyle F_{P}(x)=P((-\infty ,x])}$

is defined, and a function that is monotonically increasing and right-continuous and for which ${\displaystyle F}$

${\displaystyle \lim _{x\to -\infty }F(x)=0}$ and ${\displaystyle \lim _{x\to \infty }F(x)=1}$

applies. The first is called a distribution function of a probability distribution for the sake of differentiation , the second simply a distribution function .

From the properties of the distribution function of a probability distribution it follows directly that it is always a distribution function. The correspondence set now answers the question of whether every distribution function is always the distribution function of a probability distribution and whether the probability distribution can be reconstructed from this.

statement

Every distribution function is a distribution function of a unique probability distribution . This distribution is through ${\displaystyle F}$ ${\displaystyle P}$

${\displaystyle P_{F}((-\infty ,x]):=F(x)}$

clearly determined.

Conversely, every probability distribution determines a unique distribution function

${\displaystyle F_{P}(x):=P((-\infty ,x])}$.

Then and . ${\displaystyle F_{P_{F}}=F}$ ${\displaystyle P_{F_{P}}=P}$

The assignment of the distribution functions to the probability distributions is thus bijective .

Inferences

The correspondence theorem simplifies the investigation of probability distributions. With its help, it is often possible to dispense with methods of measurement theory, since the investigation of the distribution function using the methods of real analysis is sufficient. In addition, definitions of probability distributions can be formulated using the distribution functions. An example of this is the convergence in distribution of a random variable, which is defined by the weak convergence of distribution functions. Even far-reaching statements such as Prokhorov's theorem for probability distributions can be shown over the distribution functions. ${\displaystyle \mathbb {R} }$

In addition, by specifying a corresponding distribution function, complex probability distributions can be constructed in a targeted manner. A classic example of this is the construction of the Cantor distribution as the probability distribution with the Cantor function as the distribution function.

literature

• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 246 , doi : 10.1007 / 978-3-642-21026-6 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 70 , doi : 10.1007 / b137972 .