Some Distribution and Moment Formulae for the Markov Chain

Summary An exact formula (8) is derived for the probability distribution of an observed set of transition totals. This expression furnishes asymptotic expressions for the likelihood (24), for the covariances of transition totals (27), and for the distribution of Bartlett’s goodness-of-fit statistic (31). Formulae are also derived for the expectations of some sample functions related to the factorial moments of the transition totals (34) and for the lower moments of the estimated transition probabilities (42), (44). It is shown that the Markov chain has properties similar to those of a. set of independent multinomial distributions.