The Width of Galton- Watson Trees Conditioned by the Size

Abstract

It is proved that the moments of the width of Galton-Watson trees of size n and with offspring variance σ 2 are asymptotically given by (σ√ n) p m p where m p are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications.