  Recently, the joint probability density functions of complex eigenvalues for
products of independent complex Ginibre matrices have been explicitly derived
as determinantal point processes. We express truncated series coming from the
correlation kernels as multivariate integrals with singularity and investigate
saddle point method for such a type of integrals. As an application, we prove
that the eigenvalue correlation functions have the same scaling limits as those
of the single complex Ginibre ensemble, both in the bulk and at the edge of the
spectrum. We also prove that the similar results hold true for products of
independent truncated unitary matrices.
