Bernstein's theorem on monotone functions (original) (raw)

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Mathematical theorem

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 d n d t n f ( t ) ≥ 0 {\displaystyle (-1)^{n}{\frac {d^{n}}{dt^{n}}}f(t)\geq 0} {\displaystyle (-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 ) = ∫ 0 ∞ e − t x d g ( x ) , {\displaystyle f(t)=\int _{0}^{\infty }e^{-tx}\,dg(x),} {\displaystyle 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

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