Exact Probabilities and Asymptotic Relationships for Some Statistics from $m$-th Order Markov Chains

Abstract

An exact formula is presented for the probability of a specified frequency count of $m$-tuples $(m \geqq 1)$ in a sequence $X_1, X_2, \cdots, X_N$ from a Markov chain of order $m - 1$ having a denumerable number $a \leqq \infty$ of states. An exact expression is also obtained for the conditional probability of a specified $m$-tuple count, given the $n$-tuple count, when the chain is of order $n - 1 (n < m).$ If $a < \infty,$ then this conditional probability, when regarded as a statistic computed from the observed sequence, is shown to be asymptotically equivalent to the product of the probabilities (regarded as a statistic) associated with a corresponding set of $a^{n - 1}$ contingency tables with assigned marginals (each table having $a^{m - n}$ row and $a$ columns), where in each table the two attributes described by the table are independent. This fact leads to several simplified tests, related to standard tests of independence in contingency tables, for the null hypothesis $H_{n - 1}$ that the Markov chain is of order $n - 1$ within the alternate hypothesis $H_{m - 1}.$ Analogous results are also obtained for circular sequences.