Bernstein's theorem on monotone functions
In real analysis, a branch of mathematics, Bernstein's theorem, named after Sergei Bernstein, states that every real-valued function on the half-line that is completely monotone is a mixture of exponential functions or in more abstract language, that it is the Laplace transform of a positive Borel measure on. In one important special case the mixture is a weighted average, or expected value. It is also known as the Bernstein–Widder theorem or Hausdorff–Bernstein–Widder theorem.
History
The result was first proved by Bernstein in 1928, and similar results were discussed by David Widder in 1931 who refers to Bernstein but states that "The author had completed the proof of this theorem a few months after the publication of Bernstein's paper without being aware of its existence". The most cited reference is the 1941 book by Widder called The Laplace Transform. Later a simpler proof was given by Boris Korenblum. At around the same time Gustave Choquet studied the much more general concept of monotone functions on semigroups and gave a more abstract proof based on the Krein–Milman theorem. Felix Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem.Statement of the theorem
Complete monotonicity of a function means that is continuous on, infinitely differentiable on, and satisfiesfor all nonnegative integers and for all.The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on with cumulative distribution function such thatthe integral being a Riemann–Stieltjes integral.
Bernstein functions
Nonnegative functions whose derivative is completely monotone are called Bernstein functions. Every Bernstein function has the Lévy–Khintchine representation:where and is a measure on the positive real half-line such thatSchoenberg–Williamson theorem
The Schoenberg–Williamson theorem is the finite-order version of Bernstein's theorem. A k-monotone function satisfiesThe Schoenberg–Williamson theorem says that f is k-monotone on if and only iffor some positive measure on.The proof was published by Williamson in 1956 but in his paper he mentions that "This formula was discovered by Schoenberg in 1940 but has remained unpublished".
Using the Taylor series of with integral remainder, a more precise formula can be givenwithwhere and is the indicator function of.
Note then that if is completely monotone then it is k-monotone for all and the Post–Widder inversion formula states that converge in distribution to and converges to as goes to infinity, and we recover Bernstein's theorem.