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A note on central limit theorems for additive functionals of ergodic Markov processes

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  • Waymire, Edward C.

Abstract

This note provides a simple proof of the highly cited Kipnis-Varadhan central limit theorem for a class of (integral) additive functionals of a time-reversible, continuous parameter ergodic Markov process as a corollary to Bhattacharya’s general central limit theorem for additive functionals of ergodic Markov processes. This makes the Kipnis-Varadhan and Bhattacharya central limit theorems equivalent for the case of time-reversible Markov processes. This is revealed in a fascinating “double-limit problem” that is resolved by a simple use of Bhattacharya’s range condition on the infinitesimal generator.

Suggested Citation

  • Waymire, Edward C., 2026. "A note on central limit theorems for additive functionals of ergodic Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 200(C).
    • Handle: RePEc:eee:spapps:v:200:y:2026:i:c:s0304414926001572
    DOI: 10.1016/j.spa.2026.105025
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