  We establish a general sufficient condition for a sequence of Galton Watson
branching processes in varying environment to converge weakly. This condition
extends previous results by allowing offspring distributions to have infinite
variance, which leads to a new and subtle phenomena when the process goes
through a bottleneck and also in terms of time scales. Our assumptions are
stated in terms of pointwise convergence of a triplet of two real-valued
functions and a measure. The limiting process is characterized by a backwards
ordinary differential equation satisfied by its Laplace exponent, which
generalizes the branching equation satisfied by continuous state branching
processes. Several examples are discussed, namely branching processes in random
environment, Feller diffusion in varying environment and branching processes
with catastrophes.
