On weighted heights of random trees

Consider the family treeT of a branching process starting from a single progenitor and conditioned to havev=v(T) edges (total progeny). To each edge <e> we associate a weightW(e). The weights are i.i.d. random variables and independent ofT. The weighted height of a self-avoiding path inT starting at the root is the sum of the weights associated with the path. We are interested in the asymptotic distribution of the maximum weighted path height in the limit asv=n. Depending on the tail of the weight distribution, we obtain the limit in three cases. In particular ify²P(W(e)>y)0, then the limit distribution depends strongly on the tree and, in fact, is the distribution of the maximum of a Brownian excursion. If the tail of the weight distribution is regularly varying with exponent 0<2, then the weight swamps the tree and the answer is the asymptotic distribution of the maximum edge weight in the tree. There is a borderline case, namely,P(W(e)>y)cy^–2 asy, in which the limit distribution exists but involves both the tree and the weights in a more complicated way.