Helly-Bray's theorem

The set of Helly-Bray is a set of measure theory , a branch of mathematics that deals with the study of abstracted volume terms. These are used, for example, in stochastics or integration theory . Helly-Bray's theorem links the vague convergence of measures to the vague convergence of distribution functions and the weak convergence of measures to the weak convergence of distribution functions . It thus enables the convergence behavior of a sequence of measures to be traced back to the (point-by-point) convergence behavior of the distribution functions. The best-known example of this is the convergence in distribution in stochastics, because this is the weak convergence of probability measures and this can be attributed to the convergence of the distribution functions (in the sense of stochastics) .

The set is named after Eduard Helly and Hubert Evelyn Bray . Helly proved the theorem as early as 1912 in his work On linear functional operators , while Bray, presumably without knowing about it, published it in 1919 in his work Elementary properties of the Stieltjes integral .

Framework

On the real numbers, every finite measure defines through ${\displaystyle \mu }$

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

a so-called distribution function , which increases monotonically , is continuous and bounded on the right-hand side . Conversely, every monotonically growing right-hand side defines continuous bounded function through ${\displaystyle F}$

${\displaystyle \mu _{F}((a,b]):=F(b)-F(a)}$

a measure, the Lebesgue-Stieltjes measure . The assignment of the distribution functions to the measures is unambiguous except for one constant, that is , they generate the same measure. The question now arises how the properties of the measures are reflected in the distribution functions and vice versa. The Helly-Bray theorem makes a statement about when the convergence of the measures can be inferred from the convergence of the distribution functions. ${\displaystyle F}$ ${\displaystyle F-c}$

statement

Distribution functions are given . Then: ${\displaystyle (F_{n})_{n\in \mathbb {N} },F}$

1. If the sequence converges weakly to , then holds for every bounded continuous function ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ ${\displaystyle F}$ ${\displaystyle g}$

   ${\displaystyle \lim _{n\to \infty }\int _{\mathbb {R} }g\mathrm {d} F_{n}=\int _{\mathbb {R} }g\mathrm {d} F}$.

2. If the sequence converges vaguely to , then holds for every continuous function with compact support ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ ${\displaystyle F}$ ${\displaystyle g}$

   ${\displaystyle \lim _{n\to \infty }\int _{\mathbb {R} }g\mathrm {d} F_{n}=\int _{\mathbb {R} }g\mathrm {d} F}$.

Inferences

General

A direct conclusion from the above statements is that from the weak (vague) convergence of the distribution functions against the weak (vague) convergence of the measures against, it follows that the Stieltjes integral with respect to exactly corresponds to the integral with respect to. ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ ${\displaystyle F}$ ${\displaystyle (\mu _{F_{n}})_{n\in \mathbb {N} }}$ ${\displaystyle \mu _{F}}$ ${\displaystyle F_{n}}$ ${\displaystyle \mu _{F_{n}}}$

Finally, the converse can be shown: if the finite measures converge weakly / vague, then there exists a real sequence , so that weak / vague converges. ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle (c_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (F_{n}-c_{n})_{n\in \mathbb {N} }}$

For measures of probability

If they are all probability measures, then the sequence can be set constant to zero, since the distribution functions in the sense of probability theory are clearly defined by the conditions and . Thus the probability measures converge weakly if and only if the distribution functions converge weakly. ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle (c_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \lim _{x\to -\infty }F(x)=0}$ ${\displaystyle \lim _{x\to \infty }F(x)=1}$

In this case, caution is advised, since the weak and the vague convergence of distribution functions coincide for probability measures and the terms are not always used clearly in the literature.

Individual evidence

1. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 392.

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 387-392 , doi : 10.1007 / 978-3-540-89728-6 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 396 , doi : 10.1007 / 978-3-642-21026-6 .