  Bessel-type convolution algebras of bounded Borel measures on the matrix
cones of positive semidefinite $q\times q$-matrices over $\mathbb R, \mathbb C,
\mathbb H$ were introduced recently by R\"osler. These convolutions depend on
some continuous parameter, generate commutative hypergroup structures and have
Bessel functions of matrix argument as characters.
  Here, we first study the rich algebraic structure of these hypergroups. In
particular, the subhypergroups and automorphisms are classified, and we show
that each quotient by a subhypergroup carries a hypergroup structure of the
same type.
  The algebraic properties are partially related to properties of random walks
on matrix Bessel hypergroups. In particular, known properties of Wishart
distributions, which form Gaussian convolution semigroups on these hypergroups,
are put into a new light. Moreover, limit theorems for random walks on these
hypergroups are presented. In particular, we obtain strong laws of large
numbers and a central limit theorem with Wishart distributions as limits.
