Bernstein's theorem on monotone functions

Template:Short description In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line Template:Closed-open 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 Template:Math means that Template:Math is continuous on Template:Closed-open, infinitely differentiable on Template:Open-open, and satisfies $$(-1{)}^n\frac{d^n}{d{t}^n}f(t)\ge 0$$ for all nonnegative integers Template:Mvar and for all Template:Math. 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 Template:Closed-open with cumulative distribution function Template:Math such that $$f(t)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-tx}dg(x),$$ the integral being a Riemann–Stieltjes integral.

In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on Template:Closed-open. 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: $$f(t)=a+bt+\underset{0}{\overset{\mathrm{\infty }}{\int }}(1-e^{-tx})\mu (dx),$$ where $a,b\ge 0$ and $\mu$ is a measure on the positive real half-line such that $$\underset{0}{\overset{\mathrm{\infty }}{\int }}(1\wedge x)\mu (dx)<\mathrm{\infty }.$$

See also

• Absolutely and completely monotonic functions and sequences

References

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