Statistical Inference for Discrete Patterns

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We have studied waiting time distributions of runs and patterns in random structures such as sequences of random variables and random directed trees. The following results are derived.1. We have obtained the exact distribution of the number of "1" -runs of a specified length on {0, 1} -valued Markov trees. This result can be applied to calculate the reliability of a consecutive system on a directed tree. We also obtained the exact distribution of the life time of the consecutive system on the directed tree.2. We have studied exact distributions of sooner and later waiting times for runs in Markov dependent bivariate trials. We have given systems of linear equations with respect to conditional probability generating functions of the waiting times and have solved them. This result can be applied to calculate the reliability of the linear connected- (r, s) -out-of- (r+1, n) : F lattice system.3. We have introduced a Markov chain imbeddable vector of multinomial type and a Markov chain imbeddable variables of returnable type and have discussed some of their properties. By using the result we have derived the distribution of numbers of occurrences of runs of specified lengths in a sequence of multi-state trials.4. We have introduced a unified counting scheme for runs called 1-overlapping counting. We have given exact probability generating function of the number of 1-overlapping 1-runs of a specified length in some dependent random sequences such as a Markov chain and a heigher order Markov chain.5. We have introduced a new type of dependent sequence called a binary sequence of order (k, r) and have derived the exact distributions of sooner and later waiting times for success and failure runs in the sequence.