  The aim of this paper is to give fine asymptotics for random variables with
moments of Gamma type. Among the examples we consider are random determinants
of Laguerre and Jacobi beta ensembles with varying dimensions (the number of
observed variables and the number of measurements vary and may be different).
In addition to the Dyson threefold way of classical random matrix models (GOE,
GUE, GSE), we study random determinants of random matrices of the so-called
tenfold way, including the Bogoliubov-de Gennes and chiral ensembles from
mesoscopic physics. We show that fixed-trace matrix ensembles can be analysed
as well. Finally, we add fine asymptotics for the $p(n)$-dimensional volume of
the simplex with $p(n)+1$ points in ${\Bbb R}^n$ distributed according to
special distributions, which is strongly correlated to Gram matrix ensembles.
We use the framework of mod-$\varphi$ convergence to obtain extended limit
theorems, Berry-Esseen bounds, precise moderate deviations, large and moderate
deviation principles as well as local limit theorems. The work is especially
based on the recent work of Dal Borgo, Hovhannisyan and Rouault.
