LONG-RUN GROWTH-RATES OF DISCRETE MULTIPLICATIVE PROCESSES IN MARKOVIAN ENVIRONMENTS

If the matrix of parameters of a discrete multiplicative process is subject to certain sequentially dependent random perturbations, the long-run growth rate μ of the average process is not in general bounded above by the largest supi λi of the growth rates λi of the individual matrices which drive the process. The growth rate μ of the average process may, in general, be greater or less than the long-run growth rate λ* of a deterministic process governed by the time-averaged matrix. The time-averaged matrix may suggest that the process will be critical or subcritical (λ* ≤ 1), whereas the sequential dependence among perturbations may actually make the average process supercritical (μ > 1). This suggests that in collecting data on and analyzing the dynamics of randomly perturbed discrete multiplicative processes, it is necessary to consider possible sequential dependence among matrix parameters in addition to their relative frequencies' and average values. Applications to nuclear reactors, age-structured populations and other areas are indicated. © 1979.