Browsing by Author "Xia, Yizhou"

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    On longest consecutive patterns in Markov chains
    (2019-11-11) Xia, Yizhou; Ernst, Philip A.
    The length of longest consecutive head in Bernoulli trials L(n) has been studied extensively and has been found applications in biology, finance and non-parametric statistics. The study of longest consecutive successes in random trials dates the work of de Moivre. Limiting theorems and large deviation results are provided for L(n) with the assumption of existence of stationary distribution. Given a discrete-time homogeneous Markov chain with initial state i, one extension from previous Bernoulli case is to study the distribution of L(j,n), the length of the longest consecutive visits of this chain to state j until time n. Our work focuses on studying L(j,n) for both homogeneous and time-nonhomogeneous Markov chains. In the existing literature, no limiting theorems of L(j,n) are derived under the case of time nonhomogeneous Markov chains. We are able to solve this by first deriving a new exact formula of the distribution of L(j,n) and then derive an upper and lower bound of P(L(j,n)