Helly–Bray theorem
Encyclopedia
In probability theory
, the Helly–Bray theorem relates the weak convergence of cumulative distribution function
s to the convergence of expectation
s of certain measurable function
s. It is named after Eduard Helly
and Hubert Evelyn Bray.
Let F and F1, F2, ... be cumulative distribution functions on the real line
. The Helly–Bray theorem states that if Fn converges weakly to F, then
for each bounded
, continuous
function
g: R → R, where the integrals involved are Riemann–Stieltjes integrals.
Note that if X and X1, X2, ... are random variable
s corresponding to these distribution functions, then the Helly–Bray theorem does not imply that E(Xn) → E(X), since g(x) = x is not a bounded function.
In fact, a stronger and more general theorem holds. Let P and P1, P2, ... be probability measure
s on some set S. Then Pn converges weakly to P if and only if
for all bounded, continuous and real-valued
functions on S. (The integrals in this version of the theorem are Lebesgue–Stieltjes integrals.)
The more general theorem above is sometimes taken as defining weak convergence of measures (see Billingsley, 1999, p. 3).
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the Helly–Bray theorem relates the weak convergence of cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
s to the convergence of expectation
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
s of certain measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
s. It is named after Eduard Helly
Eduard Helly
Eduard Helly was a mathematician and the eponym of Helly's theorem, Helly families, Helly's selection theorem, Helly metric, and the Helly–Bray theorem. In 1912, Helly published a proof of Hahn–Banach theorem, 15 years before Hahn and Banach discovered it independently...
and Hubert Evelyn Bray.
Let F and F1, F2, ... be cumulative distribution functions on the real line
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
. The Helly–Bray theorem states that if Fn converges weakly to F, then
for each bounded
Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
, continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
g: R → R, where the integrals involved are Riemann–Stieltjes integrals.
Note that if X and X1, X2, ... are random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s corresponding to these distribution functions, then the Helly–Bray theorem does not imply that E(Xn) → E(X), since g(x) = x is not a bounded function.
In fact, a stronger and more general theorem holds. Let P and P1, P2, ... be probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
s on some set S. Then Pn converges weakly to P if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
for all bounded, continuous and real-valued
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
functions on S. (The integrals in this version of the theorem are Lebesgue–Stieltjes integrals.)
The more general theorem above is sometimes taken as defining weak convergence of measures (see Billingsley, 1999, p. 3).