Large deviation results are obtained for the normed limit of a supercritical multi-type branching process. Let L[i] be the normed limit of the process branching from a single individual of type i. In the case of bounded minimum growth we show P(L[i] x) = P(L[i] = 0) +x 0. In the case of exponential minimum growth we show 0. If the maximum family size is bounded then we get log P(L[i] > x) = # #.

the continuum random tree

A necessary and sufficient condition for the almost sure existence of an absolutely continuous (with respect to the branching measure) exact Hausdorff measure on the boundary of a Galton–Watson tree is obtained. In the case where the absolutely continuous exact Hausdorff measure does not exist almost surely, a criterion which classifies gauge functions φ according to whether φ-Hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given. Important examples are discussed in four additional theorems. In particular, Hawkes’s conjecture in 1981 is solved. Problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved. 1. Introduction. An