The cumulative distance from an observed to a stable age structure.

Abstract

In a closed, unisexual, age-structured population with age-specific birth and death rates which are constant in time, the vector describing a census by age categories will, as time increases, approach proportionality to the stable age structure implied by the vital rates. This stable age structure is the dominant eigenvector of a demographic projection matrix which carries out the action on a census vector of the age-specific vital rates. We show that the elementwise convergence to zero of the discounted deviations from the stable age structure is complete and exponential. The sum, over all time, of the signed discounted deviations may be easily calculated from a fundamental matrix based on the projection matrix. These results are proved for any primitive nonnegative square matrix. In the demographic context, these results suggest alternatives to an index which has been used to measure the distance from an observed to a stable age structure.