Distribution theory of runs and patterns and its applications

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In a sequence of previous projects and publications, a simple and powerful method known as "finite Markov chain imbedding" was developed to compute the exact probabilities of waiting times and occurrences for the number of words, patterns, or runs in a sequence of symbols. The technique has had a high impact in both statistics and applied probability and has been used successfully in many areas, such as reliability theory, quality control, DNA sequence analysis, psychology, astronomy, statistics, and biology. This five-year research proposal deals with further refinements to this method, and contains four parts: (i) developing a better approximation for the distribution of complex words, runs and patterns, (ii) extending Markov chain imbedding to study the boundary crossing probability for Brownian motion and its related processes, (iii) using the method to examine the distribution of clusters in DNA sequences and hospital ER wait times(a hot issue in Canada), and (iv) applying the results from (ii) to study bankruptcy and financial distress among large companies to avoid the recession as current one.