The supercritical Galton-Watson process in varying environments

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 not coincide with {Zn --> [infinity]}. Here a natural sufficient condition is given which ensures that this cannot happen. Under a weaker condition it is shown that the possible rates of growth cannot differ very much in that {Zn/EZn}1/n --> 1 on {Zn --> [infinity]}.