  We study fluctuations of linear statistics corresponding to smooth functions
for certain biorthogonal ensembles. We study those biorthogonal ensembles for
which the underlying biorthogonal family satisfies a finite term recurrence and
describe the asymptotic fluctuations using right limits of the recurrence
matrix. As a consequence, we show that whenever the right limit is a Laurent
matrix, a Central Limit Theorem holds. We will also discuss the implications
for orthogonal polynomial ensembles. In particular, we obtain a Central limit
theorem for the orthogonal polynomial ensemble associated with any measure
belonging to the Nevai Class of an interval. Our results also extend previous
results on Unitary Ensembles in the one-cut case. Finally, we will illustrate
our results by deriving Central Limit Theorems for the Hahn ensemble for
lozenge tilings of a hexagon and for the Hermitian two matrix model.
