A representation theorem and its applications to spherically-invariant random processes

The
n
th-order characteristic functions (cf) of spherically-invariant random processes (sirp) with zero means are defined as cf, which are functions of
n
th-order quadratic forms of arbitrary positive definite matrices
p
. Every
n
th-order spherically-invariant characteristic function (sicf) is represented as a weighted Lebesgue-Stieltjes integral transform of an arbitrary univariate probability distribution function
F(\cdot)
on
[0,\infty)
. Furthermore, every
n
th-order sicf has a corresponding spherically-invariant probability density (sipd). Then we show that every
n
th-order sicf (or sipd) is a random mixture of a
n
th-order Gaussian cf [or probability density]. The randomization is performed on
\nu^2 \rho
, where
\nu
is a random variable (tv) specified by the
F(\cdot)
function. Examples of sirp are given. Relations to previously known results are discussed. Various expectation properties of Gaussian random processes are valid for sirp. Related conditional expectation, mean-square estimation, semMndependence, martingale, and closure properties are given. Finally, the form of the unit threshold likelihood ratio receiver in the detection of a known deterministic signal in additive sirp noise is shown to be a correlation receiver or a matched filter. The associated false-alarm and detection probabilities are expressed in closed forms.