SMF

The Function ${\rm exp}\,{[-p\>{\rm Trace}\,\sqrt {2A}]}$ as a Laplace Transform on Symmetric Matrices

The Function ${\rm exp}\,{[-p\>{\rm Trace}\,\sqrt {2A}]}$ as a Laplace Transform on Symmetric Matrices

G. LETAC
  • Année : 1996
  • Tome : 236
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 44A10, 60E10
  • Pages : 199-213
  • DOI : 10.24033/ast.339

This note shows that if $p>0$ and if $S_+$ is the set of symmetric positive definite matrices, then the function on $S_+$ defined by $A\mapsto \exp \smash {(-{\rm Trace}\,p\sqrt {2A})}$ is the Laplace transform of a non positive function concentrated on $S_+$ if $n\ge 2$. This function is explicitely computed for $n=2$. This computation is generalized to a Lorentz cone. The link of this question with the inverse Gaussian distributions in probability theory is also discussed, as well as the general problem of considering $\det L(A)$ as a Laplace transform on symmetric matrices when $L(\lambda )$ is a Laplace transform on the real line.

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