  We describe randomized algorithms for computing the dominant eigenmodes of
the Generalized Hermitian Eigenvalue Problem (GHEP) $Ax=\lambda Bx$, with $A$
Hermitian and $B$ Hermitian and positive definite. The algorithms we describe
only require forming operations $Ax$, $Bx$ and $B^{-1}x$ and avoid forming
square-roots of $B$ (or operations of the form, $B^{1/2}x$ or $B^{-1/2}x$). We
provide a convergence analysis and a posteriori error bounds that build upon
the work
of~\cite{halko2011finding,liberty2007randomized,martinsson2011randomized}
(which have been derived for the case $B=I$). Additionally, we derive some new
results that provide insight into the accuracy of the eigenvalue calculations.
The error analysis shows that the randomized algorithm is most accurate when
the generalized singular values of $B^{-1}A$ decay rapidly. A randomized
algorithm for the Generalized Singular Value Decomposition (GSVD) is also
provided. Finally, we demonstrate the performance of our algorithm on computing
the Karhunen-Lo\`{e}ve expansion, which is a computationally intensive GHEP
problem with rapidly decaying eigenvalues.
