Characterization of exponential families of lumpable stochastic matrices

Abstract It is known that the set of lumpable Markov chains over a finite state space, with respect to a fixed lumping function, generally does not form an exponential family of stochastic matrices. In this work, we explore efficiently verifiable necessary and sufficient conditions for families of lumpable transition matrices to form exponential families. To this end, we develop a broadly applicable dimension-based method for determining whether a given family of stochastic matrices forms an exponential family.