Zhao <a href="http://www.stat.sc.edu/~ouzhao/">University of South Carolina</a> Thu, 09/30/2010 - 3:30pm In this talk I will quickly survey some recent progresses on the (conditional) central limit questions for stationary processes. One recent Markov chain example will be focused on, and its analysis is leading to some interesting phenomenon, which apparently we have not understood well yet. In vague terms, the partial sum of this Markov chain has variance growing faster than n, which delivers some challenge to prove the conditional CLT. But when we managed to show a CLT, it was found there is a `mass escaping' --- limiting variance gets smaller in some mysterious way. I will try to formulate some questions along the way. This is a joint work with D. Volny and M. Woodroofe.