An Examination of P. Davidson’s Misunderstandings about the Limiting Frequency Theory of Probability, Ergodicity and Non Ergodicity: These Approaches Require that either the Number of Observations Approaches Infinity or Time Approaches Infinity in Order for the Limit to Exist

Abstract

Paul Davidson’s technical understanding of the mathematical details of the Limiting Frequency theory of probability and Kolmogorov’s measure theoretic extension from additivity to countable additivity, which allows for an extension of the concept of the Law of Large Numbers to the concept or ergodicity and non ergodicity, where time is substituted for observations, is extremely poor. Davidson has repeatedly made errors in many journal articles published in the Journal of Post Keynesian Economics and books published by Edward Elgar about these concepts since 1979. For instance, one example of a fundamental error made by Davidson since 1982 is his assertion, contradicted by all mathematical statisticians, that “Non stationarity is a sufficient, but not a necessary, condition for nonergodicity”. Davidson has repeated this false claim many times in the literature since 1982 when it first appeared in an article published in the Journal of Post Keynesian Economics. It is very simple to grasp what Davidson’s fundamental error is from just the title of a 1973 paper published in The Annals of Probability by R W Madsen and D L Isaacson, titled “Strongly Ergodic Behavior for Non-Stationary Markov Processes”, to realize that the correct mathematical statement is that non-stationary processes can be ergodic. This means that the correct analysis, known for certain by all mathematical statisticians since 1973, is that “Non stationarity is a necessary, but not a sufficient, condition for nonergodicity”. There is no existing evidence that Paul Davidson had any knowledge of Keynes’s approach to measurement, which rejected the exact and precise approach of Kolmogorov and Tinbergen except as a very special case, in favor of an inexact approach to measurement based on approximation and imprecise, interval valued probability and outcomes. Given that the concepts of ergodicity and non ergodicity require as a necessary condition the acceptance of the measure theoretic approach of Kolmogorov,it is straightforward to conclude that Keynes would reject the concepts of ergodicity and non ergodicity in macroeconomics or in regard to the concept of uncertainty because Part II of the TP establishes non additivity and non linearity as the general case and additivity and linearity as a special case, which is what the Kolmogorov axioms require.