A natural sufficient condition is given for a Galton-Watson process in a varying environment to have a single rate of growth that obtains throughout the survival set of the process. In the homogeneous process the growth rate is provided by the usual Seneta-Heyde norming.

Let {Zn} be a supercritical Galton-Watson process in varying environments. It is known that Zn when normed by its mean EZn converges almost surely to a finite random variable W. It is possible, however, for such a process to exhibit more than one rate of growth so that in particular {W 0} need...