The value at the mode in multivariate $t$ distributions: a curiosity or not?

TL;DR

This paper investigates the behavior of the mode value in multivariate Student's t distributions as degrees of freedom increase, revealing a dimension-dependent monotonicity change that raises questions about its significance.

Contribution

It proves that the mode value's convergence to the Gaussian case varies with dimension, changing from increasing to decreasing as dimension increases beyond two.

Findings

Mode value converges monotonically to Gaussian value as degrees of freedom increase.

Monotonicity is dimension-dependent: increasing in 1D, constant in 2D, decreasing in 3D and higher.

Raises questions about the significance of this dimension-dependent behavior.

Abstract

It is a well-known fact that multivariate Student $t$ distributions converge to multivariate Gaussian distributions as the number of degrees of freedom $\nu$ tends to infinity, irrespective of the dimension $k\geq1$. In particular, the Student's value at the mode (that is, the normalizing constant obtained by evaluating the density at the center) $c_{\nu,k}=\frac{\Gamma(\frac{\nu+k}{2})}{(\pi \nu)^{k/2} \Gamma( \frac{\nu}{2})}$ converges towards the Gaussian value at the mode $c_k=\frac{1}{(2\pi)^{k/2}}$. In this note, we prove a curious fact: $c_{\nu,k}$ tends monotonically to $c_k$ for each $k$, but the monotonicity changes from increasing in dimension $k=1$ to decreasing in dimensions $k\geq3$ whilst being constant in dimension $k=2$. A brief discussion raises the question whether this \emph{a priori} curious finding is a curiosity, \emph{in fine}.