Editing Helly–Bray theorem

In [[probability theory]], the '''Helly–Bray theorem''' relates the [[convergence in distribution|weak convergence]] of [[cumulative distribution function]]s to the convergence of [[expected value|expectation]]s of certain [[measurable function]]s.  It is named after [[Eduard Helly]] and [[Hubert Evelyn Bray]].

Let ''F'' and ''F''<sub>1</sub>, ''F''<sub>2</sub>, ... be cumulative distribution functions on the [[real number|real line]]. The Helly–Bray theorem states that if ''F''<sub>''n''</sub> converges weakly to ''F'', then

::<math>\int_\mathbb{R} g(x)\,dF_n(x) \quad\xrightarrow[n\to\infty]{}\quad \int_\mathbb{R} g(x)\,dF(x)</math>

for each [[bounded function|bounded]], [[continuous function]] ''g'': '''R''' &rarr; '''R''', where the integrals involved are [[Riemann&ndash;Stieltjes integral]]s.

Note that if ''X'' and ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... are [[random variable]]s corresponding to these distribution functions, then the Helly–Bray theorem does not imply that E(''X''<sub>''n''</sub>) &rarr; E(''X''), since ''g''(''x'') = ''x'' is not a bounded function.

In fact, a stronger and more general theorem holds.  Let ''P'' and ''P''<sub>1</sub>, ''P''<sub>2</sub>, ... be [[probability measure]]s on some [[Set (mathematics)|set]] ''S''.  Then ''P''<sub>''n''</sub> converges weakly to ''P'' [[if and only if]]

::<math>\int_S g \,dP_n \quad\xrightarrow[n\to\infty]{}\quad \int_S g \,dP,</math>

for all bounded, continuous and [[real number|real-valued]] functions on ''S''.  (The integrals in this version of the theorem are [[Lebesgue–Stieltjes integral]]s.)

The more general theorem above is sometimes taken as ''defining'' [[weak convergence of probability measures|weak convergence of measures]]  (see Billingsley, 1999, p.&nbsp;3).

==References==
#{{cite book | author=Patrick Billingsley | authorlink=Patrick Billingsley | title=Convergence of Probability Measures, 2nd ed. | publisher=John Wiley & Sons, New York | year=1999 | isbn=0-471-19745-9 | url-access=registration | url=https://archive.org/details/convergenceofpro0000bill }}

{{PlanetMath attribution|id=3710|title=Helly–Bray theorem}}

{{DEFAULTSORT:Helly-Bray theorem}}
[[Category:Theorems in probability theory]]