  We obtain an optimal exponential square integrability theorem for the Bergman
projection of a function bounded by 1 in modulus. This is interpreted as the
statement that the asymptotic tail variance of such a function is at most 1.
The asymptotic tail variance defines a seminorm on the Bloch space. We apply
the main result to quasiconformal Teichm\"uller theory, and obtain an estimate
of the integral means spectrum of k-quasiconformal mappings that are conformal
in the exterior disk: $B(k,t)\le\frac14k^2|t|^2(1+7k)^2$. This is conjectured
asymptotically sharp as k tends to 0, by Prause and Smirnov (2011).
