Ed Waymire: On Central Limit Theorems & Law of the Iterated Logarithm for Additive Functionals of Ergodic Markov Processes

A new and simple proof of the Kipnis-Varadhan central limit theorem will be presented for additive functionals of a 

time-reversible continuous parameter Markov process based on Bhattacharya’s more general central limit theorem for

 ergodic Markov processes. As a corollary this proof makes Bhattacharya’s central limit theorem 

and the Kipnis-Varadhan central limit theorem equivalent for the case of time-reversible Markov 

processes.  In this regard, a fascinating “double-limit problem” is resolved by a simple use of 

Bhattacharya’s range condition on the infinitesimal generator.  A simple extension for the time-reversible

case of the Strassen-type law of the iterated logarithm, more generally obtained by Bhattacharya, also follows 

by the approach presented here.   Bhattacharya’s central limit theorem and that of Kipnis-Varadhan have each

each proved to be valuable in applications to dispersion of solutes  and to interacting particle systems, 

respectively.