The Ergodic Theorems of Demography: a Simple Proof

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.

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Arthur, W.B. The Ergodic Theorems of Demography: a Simple Proof. Demography 19, 439–445 (1982). https://doi.org/10.2307/2061011

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Keywords

  • Time Zero
  • Ergodic Theorem
  • Vital Rate
  • Vital Event
  • Smoothing Process