  We prove a conjecture of Lalley and Sellke [Ann. Probab. 15 (1987)] asserting
that the empirical (time-averaged) distribution function of the maximum of
branching Brownian motion converges almost surely to a double exponential, or
Gumbel, distribution with a random shift. The method of proof is based on the
decorrelation of the maximal displacements for appropriate time scales. A
crucial input is the localization of the paths of particles close to the
maximum that was previously established by the authors [Comm. Pure Appl. Math.
64 (2011)].
