Operator-Theoretical Treatment of Ergodic Theorem and Its Application to Dynamic Models in Economics

Abstract

The purpose of this paper is to study the time average behavior of Markov chains with transition probabilities being kernels of completely continuous operators, and therefore to provide a sufficient condition for a class of Markov chains that are frequently used in dynamic economic models to be ergodic. The paper reviews the time average convergence of the quasi-weakly complete continuity Markov operators to a unique projection operator. Also, it shows that a further assumption of quasi-strongly complete continuity reduces the dependence of the unique invariant measure on its corresponding initial distribution through ergodic decomposition, and therefore guarantees the Markov chain to be ergodic up to multiplication of constant coefficients. Moreover, a sufficient and practical condition is provided for the ergodicity in economic state Markov chains that are induced by exogenous random shocks and a correspondence between the exogenous space and the state space.