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

It is a well-known fact that multivariate Student $t$ distributions converge to multivariate Gaussian distributions as the number of degrees of freedom $ν$ 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) $c_{ν,k}=\frac{Γ(\frac{ν+k}{2})}{(πν)^{k/2} Γ( \fracν{2})}$ converges towards the Gaussian value at the mode $c_k=\frac{1}{(2π)^{k/2}}$. In this note, we prove a curious fact: $c_{ν,k}$ 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}.

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