Reforming the Wishart characteristic function

TL;DR

This paper clarifies the analytic structure of the Wishart distribution's characteristic function, revealing limitations for matrices larger than 2x2 and providing a corrected extension using the Fourier-Laplace transform of Wishart processes.

Contribution

It demonstrates the ambiguity in the characteristic function for matrices larger than 2x2 and offers a new analytic extension via the Fourier-Laplace transform.

Findings

The characteristic function is unambiguous only for 2x2 matrices.

For larger matrices, the determinant's complex range causes ambiguity.

A corrected analytic extension is provided using the Fourier-Laplace transform.

Abstract

The literature presents the characteristic function of the Wishart distribution on m times m matrices as an inverse power of the determinant of the Fourier variable, the exponent being the positive, real shape parameter. I demonstrate that only for two times two matrices, this expression is unambiguous -- in this case the complex range of the determinant excludes the negative real line. When m greater or equals 3 the range of the determinant contains closed lines around the origin, hence a single branch of the complex logarithm does not suffice to define the determinant's power. To resolve this issue, I give the correct analytic extension of the Laplace transform, by exploiting the Fourier-Laplace transform of a Wishart process.