  In this note, we show that a central limit theorem holds for exterior powers
of the Kontsevich-Zorich (KZ) cocycle. In particular, we show that, under the
hypothesis that the top Lyapunov exponent on the exterior power is simple, a
central limit theorem holds for the lift of the (leafwise) hyperbolic Brownian
motion to any strongly irreducible, symplectic,
$\text{SL}(2,\mathbb{R})$-invariant subbundle, that is moreover
symplectic-orthogonal to the so-called tautological subbundle. We then show
that this implies that a central limit theorem holds for the lift of the
Teichm\"uller geodesic flow to the same bundle.
  For the random cocycle over the hyperbolic Brownian motion, we are able to
prove under the same hypotheses that the variance of the top exponent is
strictly positive. For the deterministic cocycle over the Teichm\"uller
geodesic flow we can prove that the variance is strictly positive only for the
top exponent of the first exterior power (the KZ cocycle itself) under the
hypothesis that its Lyapunov spectrum is simple.
