  We consider random walks on the cone of $m \times m$ positive definite
matrices, where the underlying random matrices have orthogonally invariant
distributions on the cone and the Riemannian metric is the measure of distance
on the cone. By applying results of Khare and Rajaratnam (Ann. Probab., 45
(2017), 4101--4111), we obtain inequalities of Hoffmann-J{\o}rgensen type for
such random walks on the cone. In the case of the Wishart distribution
$W_m(a,I_m)$, with index parameter $a$ and matrix parameter $I_m$, the identity
matrix, we derive explicit and computable bounds for each term appearing in the
Hoffmann-J{\o}rgensen inequalities.
