Characteristic Function (probability Theory) - Generalizations

Generalizations

The notion of characteristic functions generalizes to multivariate random variables and more complicated random elements. The argument of the characteristic function will always belong to the continuous dual of the space where random variable X takes values. For common cases such definitions are listed below:

• If X is a k-dimensional random vector, then for t ∈ Rk

• If X is a k×p-dimensional random matrix, then for t ∈ Rk×p

• If X is a complex random variable, then for t ∈ C

• If X is a k-dimensional complex random vector, then for t ∈ Ck

• If X(s) is a stochastic process, then for all functions t(s) such that the integral ∫_Rt(s)X(s)ds converges for almost all realizations of X

Here denotes matrix transpose, tr(·) — the matrix trace operator, Re(·) is the real part of a complex number, z denotes complex conjugate, and * is conjugate transpose (that is z* = zT ).