  The random variable $1+z_1+z_1z_2+\dots$ appears in many contexts and was
shown by Kesten to exhibit a heavy tail distribution. We consider natural
extensions of this variable and its associated recursion to $N \times N$
matrices either real symmetric $\beta=1$ or complex Hermitian $\beta=2$. In the
continuum limit of this recursion, we show that the matrix distribution
converges to the one of the inverse-Wishart ensemble of random matrices. The
full dynamics is solved using a mapping to $N$ fermions in a Morse potential,
which are non-interacting for $\beta=2$. At finite $N$ the distribution of
eigenvalues exhibits heavy tails, generalizing Kesten's results in the scalar
case. The density of fermions in this potential is studied for large $N$, and
the power-law tail of the eigenvalue distribution is related to the properties
of the so-called determinantal Bessel process which describes the hard edge
universality of random matrices. For the discrete matrix recursion, using free
probability in the large $N$ limit, we obtain a self-consistent equation for
the stationary distribution. The relation of our results to recent works of
Rider and Valk\'o, Grabsch and Texier, as well as Ossipov, is discussed.
