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I'm a math grad who is interested in learning more about economics for fun.

Reading through RMT, I saw some interesting math (in chapter 2) around using "invariant functions" to determine the ergodicity of a matrix. After a google/bing search the only mention of "invariant functions" were also referring to RMT.

Another interesting statement (stated without proof) is that ergodic probability measures on Markov chains are always extremal. Another result that did not seem to come up in google or my own references.

Since I have never seen these types of results, despite their apparent simplicity and utility, I was wondering if I am missing a reference. Does anyone know a good reference that might go through some of these types of results? RMT states them without proof.

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    $\begingroup$ I guess they come from more sophisticated sources by way of specialization. After all, they use invariant functions to define ergodicity. The extremality of ergodic measures on Markov chains seems to come from the ergodic decomposition, which can be proven using extreme point methods by Choquet's theorem. $\endgroup$ Commented Jan 23, 2023 at 7:00

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