Long-Run Growth Rates of Discrete Multiplicative Processes in Markovian Environments

If the matrix oi pararrreters of a discrete multiplicative process 1s suhecl Cu certain sequentially dependent random perturbations, the long-run growth rate of the average process is not in general bounded above by the largest sup, Ai of the growth rates A, of the individual matrices which drive the process. The growti rate p of the average process may, in general, be greater or less than the long-rur growth rate A* of a deterministic process governed by the time-averaged matrix. The time-averaged matrix may suggest that the process wili be critical or subcritical (A* < I), whereas the sequential dependence among perturbations may actually make the average process supercritical (p > 1). This suggeata that in collecting data on and analyzing the dynamics of randomly perturbed discrete multiplicative procesaea, it is necessary to consider possible sequential dependence among matrix parameters in addition to their relative frequencies and zverage values. Applications to nuclear reactors, age-structured populationa and other areas are indicated.