  We introduce a notion of Gieseker stability for a filtered holomorphic vector
bundle $F$ over a projective manifold. We relate it to an analytic condition in
terms of hermitian metrics on $F$ coming from a construction of the Geometric
Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K.
Donaldson. We prove that if there is a $\tau$-Hermite-Einstein metric $h_{HE}$
on $F$, then there exists a sequence of such balanced metrics that converges
and its limit is $h_{HE}$. As a corollary, we obtain an approximation theorem
for coupled Vortex equations that cover in particular the cases of
Hermite-Einstein equations, Garcia-Prada and Bradlows's coupled Vortex
equations and special Vafa-Witten equations.
