# Bernstein's theorem on monotone functions

In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average, or expected value.

Total monotonicity (sometimes also complete monotonicity) of a function f means that f is continuous on [0, ∞), infinitely differentiable on (0, ∞), and satisfies

$(-1)^{n}{\frac {d^{n}}{dt^{n}}}f(t)\geq 0$ for all nonnegative integers n and for all t > 0. Another convention puts the opposite inequality in the above definition.

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

$f(t)=\int _{0}^{\infty }e^{-tx}\,dg(x),$ the integral being a Riemann–Stieltjes integral.

In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on [0, ∞). In this form it is known as the Bernstein–Widder theorem, or Hausdorff–Bernstein–Widder theorem. Felix Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem.

## Bernstein functions

Nonnegative functions whose derivative is completely monotone are called Bernstein functions. Every Bernstein function has the Lévy–Khintchine representation:

$f(t)=a+bt+\int _{0}^{\infty }(1-e^{-tx})\,\mu (dx),$ where $a,b\geq 0$ and $\mu$ is a measure on the positive real half-line such that

$\int _{0}^{\infty }(1\wedge x)\,\mu (dx)<\infty .$ This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.