Bernstein's theorem on monotone functions

The result was first proved by Bernstein in 1928,^[1] and similar results were discussed by David Widder in 1931^[2] 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.^[3][4] Later a simpler proof^[5][6][7] 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^{[8][9][10][11][12]} based on the Krein–Milman theorem. Felix Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem.

Complete monotonicity of a function f means that f is continuous on [0, ∞), infinitely differentiable on (0, ∞), and satisfies$${\displaystyle (-1)^{n}{\frac {d^{n}}{dt^{n}}}f(t)\geq 0}$$for all nonnegative integers n and for all t > 0.

The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on [0, ∞) with cumulative distribution function g such that$${\displaystyle f(t)=\int _{0}^{\infty }e^{-tx}\,dg(x),}$$the integral being a Riemann–Stieltjes integral.

Nonnegative functions whose derivative is completely monotone are called Bernstein functions.^[11] Every Bernstein function has the Lévy–Khintchine representation:$${\displaystyle f(t)=a+bt+\int _{0}^{\infty }\left(1-e^{-tx}\right)\mu (dx),}$$where ${\displaystyle a,b\geq 0}$ and ${\displaystyle \mu }$ is a measure on the positive real half-line such that$${\displaystyle \int _{0}^{\infty }\left(1\wedge x\right)\mu (dx)<\infty .}$$

The Schoenberg–Williamson theorem (also called Schoenberg's theorem on multiply monotone functions, Williamson's representation theorem) is the finite-order version of Bernstein's theorem. A k-monotone (or k-times monotone) function satisfies$${\displaystyle (-1)^{n}{\frac {d^{n}}{dt^{n}}}f(t)\geq 0\quad {\text{for}}\,\,\,0\leq n\leq k}$$The Schoenberg–Williamson theorem says that f is k-monotone on (0, ∞) if and only if$${\displaystyle f(t)=\int _{0}^{\infty }(x-t)_{+}^{k-1}\,d\mu (x)}$$for some positive measure ${\displaystyle \mu }$ on (0, ∞).

The proof was published by Williamson in 1956^[13] but in his paper he mentions that "This formula was discovered by Schoenberg in 1940 but has remained unpublished".

Using the Taylor series of ${\displaystyle f}$ with integral remainder, a more precise formula can be given^[14]$${\displaystyle f(t)=\int _{0}^{\infty }\left(1-{\frac {tx}{n}}\right)_{+}^{n-1}f_{n}(x)\,dx}$$with$${\displaystyle f_{n}(x)={\frac {(-1)^{n}}{n!}}\left({\frac {n}{x}}\right)^{n+1}f^{(n)}\left({\frac {n}{x}}\right)I(x>0)}$$where ${\displaystyle f^{(n)}(t)={\frac {d^{n}}{dt^{n}}}f(t)}$ and ${\displaystyle I(x>0)}$ is the indicator function of ${\displaystyle \mathbb {R} _{+}^{*}}$.

Note then that if ${\displaystyle f}$ is completely monotone then it is k-monotone for all ${\displaystyle k\in \mathbb {N} }$ and the Post–Widder inversion formula states that ${\displaystyle f_{n}}$converge in distribution to ${\displaystyle \mu }$ and ${\displaystyle \left(1-{\frac {tx}{n}}\right)_{+}^{n-1}}$ converges to ${\displaystyle e^{-tx}}$as ${\displaystyle n}$ goes to infinity, and we recover Bernstein's theorem.^[14]

• Absolutely and completely monotonic functions and sequences
• Choquet's theorem
• Krein–Milman theorem