Projection of mean and variance functions for population processes with time-homogeneous laws of evolution

This paper provides algorithms for projection of mean and covariance functions for stochastic population processes governed by time-homogeneous laws of fertility and mortality. The theoretical foundation of the algorithms is general age-dependent branching processes in discrete time. The algorithms are employed in several illustrative projections, based on 1982 Chinese data, of a population experiencing an abrupt transition to below replacement fertility. Methods of constructing confidence limits for total population size are illustrated. Also developed are procedures for projecting mean and variance functions for populations which may be heterogeneous with respect to mortality or fertility. The projections performed yield two observations. First, the coefficient of variation in population size appears to be inversely related to the Malthusian parameter of population growth. Second, the coefficient of variation for population size is negligible for large homogeneous initial populations. But when the initial population is heterogeneous with respect to fertility or mortality, then substantial coefficients of variation, exceeding 0.4, are observed in some of the projections performed.