  We consider random analytic functions given by a Taylor series with
independent, centered complex Gaussian coefficients. We give a new sufficient
condition for such a function to have bounded mean oscillations. Under a mild
regularity assumption this condition is optimal. Using a theorem of Holland and
Walsh, we give as a corollary a new bound for the norm of a random Gaussian
Hankel matrix. Finally, we construct some exceptional Gaussian analytic
functions which in particular disprove the conjecture that a random analytic
function with bounded mean oscillations always has vanishing mean oscillations.
