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}.
Categories: math.ST math.PR stat.TH