The central limit theorem for Markov chains with normal transition operators, started at a point

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Central limit theorems for additive functionals of ergodic Markov diffusions processes (2011, background): “For non-reversible Markov chains, a systematic study of fractional Poisson equation is done in Derriennic and Lin (2001b). The connection with the rate of convergence of Ptf is also discussed therein, and the result “at equilibrium” is extended to an initial δx Dirac mass in Derriennic and Lin (2001”
On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria (2007, background): “…condition 1. is just the Cauchy criterion for the convergence of =1 P 0 ( X i ) in L 2 , and the condition 2. means exactly that The conditions 1. and 2. are given in Theorem C of Derriennic and Lin (2001) and are due to Gordin and Lifshitz (see the discussion on page 511 in Derriennic and Lin).”
Central limit theorem for Markov processes with spectral gap in the Wasserstein metric (2011, unknown): “[32] for quasi-reversible, [9, 17] for normal processes.”
Pointwise ergodic theorems with rate and application to limit theorems for stationary processes (2009, methodology): “In particular, we will be interested in the Law of the Iterated Logarithm or,in the case where {Xn} is given by the functional of a Markov chain, in the quenched Central Limit Theorem.”