The Ergodic Theorems of Demography: a Simple Proof


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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  1. Arthur, W. B. 1981.Why a Population Converges to Stability. American Mathematical Monthly 88:557–563.

    Article  Google Scholar 

  2. Coale, A. J. 1957. How the Age Distribution of a Human Population is Determined. Cold Spring Harbor Symposia in Quantitative Biology 22: 83–89.

    Google Scholar 

  3. Coale, A. J. 1972. The Growth and Structure of Human Populations. Princeton, N.J.: Princeton University Press.

    Google Scholar 

  4. Cohen, J. E. 1979. Ergodic Theorems in Demography. Bulletin of the American Mathematical Society 1:275–295.

    Article  Google Scholar 

  5. Feller, W. 1968. Introduction to Probability Theory and its Applications. Vol. 1, 3rd Ed. New York: Wiley.

    Google Scholar 

  6. Kim, Y. J., and Z. M. Sykes. 1976. An Experimental Study of Weak Ergodicity in Human Populations. Theoretical Population Biology 10:150–172.

    Article  Google Scholar 

  7. Leslie, P. H. 1945. On the Use of Matrices in Certain Population Mathematics. Biometrika 33: 183–212.

    Article  Google Scholar 

  8. Lopez, A. 1961. Problems in Stable Population Theory. Princeton, N.J.: Office of Population Research.

    Google Scholar 

  9. Lopez, A. 1967. Asymptotic Properties of a Human Age Distribution Under a Continuous Net Maternity Function. Demography 4:680–687.

    Article  Google Scholar 

  10. Lotka, A. J. and Sharpe, F. R. 1911. A Problem in Age Distribution. Philosophical Magazine, Ser. 6,21:339–345.

    Google Scholar 

  11. McFarland, D. D. 1969. On the Theory of Stable Populations: A New and Elementary Proof of the Theorems Under Weaker Assumptions. Demography 6:301–322.

    Article  Google Scholar 

  12. Parlett, B. 1970. Ergodic Properties of Populations I: The One Sex Model. Theoretical Population Biology 1:191–207.

    Article  Google Scholar 

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Arthur, W.B. The Ergodic Theorems of Demography: a Simple Proof. Demography 19, 439–445 (1982).

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