Permanental Vectors

TL;DR

This paper characterizes all permanental vectors in three dimensions, expanding understanding of their structure and providing applications to higher dimensions and permanental processes.

Contribution

It offers a complete characterization of three-dimensional permanental vectors and explores their applications in higher dimensions and related stochastic processes.

Findings

Characterization of all 3D permanental vectors

Extension to applications in higher dimensions

Insights into permanental processes

Abstract

A permanental vector is a generalization of a vector with components that are squares of the components of a Gaussian vector, in the sense that the matrix that appears in the Laplace transform of the vector of Gaussian squares is not required to be either symmetric or positive definite. In addition the power of the determinant in the Laplace transform of the vector of Gaussian squares, which is -1/2, is allowed to be any number less than zero. It was not at all clear what vectors are permanental vectors. In this paper we characterize all permanental vectors in $R^{3}_{+}$ and give applications to permanental vectors in $R^{n}_{+}$ and to the study of permanental processes.