The Ergodic Theorems of Demography: a Simple Proof

Abstract

Abstract Standard proofs of the ergodic theorems of demography rely on theorems borrowed from positive matrix theory, tauberian theory, and the theory of time-inhomogeneous Markov matrices. These proofs are efficient and expedient, but they give little direct insight into the mechanism that causes ergodicity. This paper proposes a simple and unified proof of the two ergodic theorems. It is shown that the birth dynamics can be decomposed into a smoothing process that progressively levels out past fluctuations in the birth sequence and a reshaping process that accounts for current period-to-period changes in vital rates. The smoothing process, which causes the birth sequence to lose information on its past shape, is shown to be the ergodic mechanism behind both theorems.