Geometric growth for stochastic difference equations with application to branching populations (Q882889)

The article considers the asymptotic growth rate for an integer-valued random process given by a stochastic difference equation driven by a martingale difference sequence. Under certain technical conditions, a suitably normalized version of this process is shown to converge in \(\alpha\)th mean for a certain \(\alpha\in[1, 2]\) to a nonnegative random variable, which is strictly positive if and only if the process under consideration explodes. Under an additional condition, this in turn leads to a result that implies in particular geometric growth on the explosion set. The results are applied to asexual controlled branching processes with random control functions and bisexual branching processes.