# Bernstein's theorem on monotone functions

(Redirected from Completely monotone function)

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

${\displaystyle (-1)^{n}{{\rm {d}}^{n} \over {\rm {d}}t^{n}}f(t)\geq 0}$

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

${\displaystyle f(t)=\int _{0}^{\infty }{\rm {e}}^{-tx}\,{\rm {d}}g(x),}$

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:

${\displaystyle f(t)=a+bt+\int _{0}^{\infty }(1-{\rm {e}}^{-tx})\mu ({\rm {d}}x)}$

where ${\displaystyle a,b\geq 0}$ and ${\displaystyle \mu }$ is a measure on the positive real half-line such that

${\displaystyle \int _{0}^{\infty }(1\wedge x)\mu ({\rm {d}}x)<\infty .}$

## References

• S. N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica. 52: 1–66. doi:10.1007/BF02592679.
• D. Widder (1941). The Laplace Transform. Princeton University Press.
• Rene Schilling, Renming Song and Zoran Vondracek (2010). Bernstein functions. De Gruyter.