Existence of critical tiltings and local limits of general size-conditioned Bienaymé-Galton-Watson multitype trees

Abstract

We are interested in the structure of multitype Bienaymé-Galton-Watson (BGW) trees conditioned on integer linear combinations of the numbers of vertices of given types. We show that, under regularity assumptions on the offspring distributions, it is always possible to find a critical BGW tree having the same conditional distribution. This allows us to prove the existence of local limits for noncritical BGW trees, under a large variety of conditionings. Our proof is based on geometric considerations on the set of the so-called exponential tiltings of a family of offspring distributions.

28 pages, 1 figure