  We establish the asymptotic normality of the dimension of large-size random
Fishburn matrices by a complex-analytic approach. The corresponding dual
problem of size distribution under large dimension is also addressed and
follows a quadratic type normal limit law. These results represent the first of
their kind and solve two open questions raised in the combinatorial literature.
They are presented in a general framework where the entries of the Fishburn
matrices are not limited to binary or nonnegative integers. The analytic
saddle-point approach we apply, based on a powerful transformation for
$q$-series due to Andrews and Jel\'inek, is also useful in solving a conjecture
of Stoimenow in Vassiliev invariants.
