Degenerate distribution

in an $n$-dimensional Euclidean space

Any probability distribution having support on some (linear) manifold of dimension smaller than $n$. Otherwise the distribution is called non-degenerate. A degenerate distribution in the case of finite second moments is characterized by the fact that the rank $r$ of the corresponding covariance (or correlation) matrix is smaller than $n$. Here $r$ coincides with the smallest dimension of the linear manifolds on which the given degenerate distribution is supported. The concept of a degenerate distribution can be clearly extended to distributions in linear spaces. The name improper distributions is sometimes given to degenerate distributions, while non-degenerate distributions are sometimes called proper distributions.

Comments

An improper distribution also refers, in Bayesian statistics (cf. Bayesian approach; Bayesian approach, empirical), to a measure of infinite total mass, which is still being manipulated as a probability distribution. Cf. also Improper distribution.