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

2012/11/06 by Christophe Ley, Ley, Christophe, Anouk Neven +1 · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST) #math.PR #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.1211.1174

8 pages, 2 figures, 1 table

arxiv created 2014/01/10 · arxiv updated 2014/01/13

Abstract

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≥1. 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=\fracΓ((ν+k)/(2))(πν)^k/2 Γ( \fracν2) converges towards the Gaussian value at the mode c_k=\frac1(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≥3 whilst being constant in dimension k=2. A brief discussion raises the question whether this a priori curious finding is a curiosity, in fine.

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