On longest consecutive patterns in Markov chains
Ernst, Philip A.
Doctor of Philosophy
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)<k) without the assumption of the existence of stationary distribution. Then we offer one convergence in probability theorem and one convergence almost surely theorem of L(j,n). We also offer a limiting result respect to the expectation of L(j,n). We also close an open problem about the large deviation results of L(j,n). We first establish asymptotics for the moment generating function of L(j,n) in general Markov chains without the assumption of the existence of stationary distribution and then we provide two large deviation principles of L(j,n). The existing large deviation results only consider Bernoulli trials and two state homogeneous Markov chains. Last, we provide two extensions. One is to study the length of longest consecutive visit of a pattern in Markov chains and the other one is to study the length of longest consecutive visit of a set of states in Markov chains. We consider Matrix to represent probabilities and theorems of primitive matrix help us to prove limiting theorems. Simulation results are also included.
Markov chain; Discrete stochastic process; Large deviation principle; Law of large number