A semigroup proof of a non-ergodic Markov process central limit theorem with third order terms (Q1279804)

A semigroup product formula for linear operators, \[ \begin{multlined} T\Biggl({u\over n}\Biggr)^n f(x)=\\ \sum^M_{m=1} \Biggl(A^m_0+ {u\over n} A^m_1+ {u^2\over n^2} A^m_2\Biggr) \exp\Biggl(in\theta_m+ uB^m_0+ {u^2\over n} B^m_1+ {u^3\over n^2} B^m_2\Biggr) f+ O(n^{-3} p^4\tau(n, t))\end{multlined} \] is derived and used to prove a central limit theorem with third-order terms applying to all ergodic and non-ergodic discrete time Markov processes with continuous transition probability densities on a compact Hausdorff space.