2019/01/27 by Mayerhofer, Eberhard
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1901.09347
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.