Exponential Family of Markov Kernels and Asymptotic Exponential Family of Markov Sources

摘要

For parametric models of Markov sources, we prove that the notion of asymptotic exponential family is equivalent to the notion of exponential family of Markov kernels. The former was introduced by Takeuchi and Barron (1998) for families of general stochastic processes, inspired by the exponential family of Markov chains discovered by Ito and Amari (1988), while the latter was introduced by Nakagawa and Kanaya (1993) for one-dimensional families of Markov sources based on the discussion by Ito and Amari (1988) and later the general form was established by Nagaoka (2005). The discussion in this report is some refinement of works of Takeuchi and Nagaoka (2017).