Section 5 Maxwell’s distribution

5.1

Density: Radon-Nikodym derivative

5.5

Bochner’s theorem for characteristic functions

A function g is the characteristic function of a probability density distribution function f, if and only if g satisfies

(1) g is continuous,

(2) g(0) = 1

(3) g is positive definite: that is, for any positive integer n and complex numbers 

 {c_k} (k = 1, ..., n) \sum_{i,j} c_i^* g(z_i - z_j) c_j >= 0. 

This theorem may be used, e.g., to model autocorrelation functions in nonequilibrium statistical mechanics.

Proof: A proof may found in, e.g., 

K Yosita, Functional Analysis (6th ed) (Springer 1980) p346

W Feller, An introduction of probability theory and its applications (Wiley, 1966, 1971) Vol. II Section XIX 2 p620-.