Finite Markov Chain Imbedding and Its Applications in Stochastic Processes, biological Sequences, and Discrete Mathematics

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The finite Markov chain imbedding (FMCI) technique is an unconventional, simple, flexible and computation efficient probabilistic tool to evaluate probabilities and distributions of runs and patterns of interest. It has been successfully applied for solving complex and unsolved problems in various areas such as health science, genomic analysis, reliability, quality control, physics, statistics, applied probability, computer science, and discrete mathematics. In this proposal, the FMCI technique is going to be extended into four very important applied areas, (i) boundary crossing probability (BCP) for high dimensional Brownian motion, (ii) matching probability of two DNA sequences with allowing at most d mutations, (iii) distributions of patterns to avoid in [S]-specified random permutation and (iv) distributions of bumps of genome-wide association studies for comparing gene expressions between normal and disease chromosomes. The following are expected results:  ***A. Short term expected results (next five years)***(i)   The first part of the proposal will establish an analytical and efficient numerical method for approximating the boundary crossing probabilities for non-linear convex boundaries of d-dimensional Brownian motion. It will show the rate of convergence is O(1/n1/2) and independent of the dimensionality d. The results will be extended to related stochastic processes such as Ornstein-Uhlenbeck process and Brownian Bridge.***(ii)  The statistic Ln(d), the length of the longest matching of two DNA sequences with allowing at most d mutations/insertions, is proposed as a measure for similarity. The exact distributions of Ln(d) will be derived and show that the distribution of Ln(d) can be expressed in terms of the distribution of scan statistics.***(iii)  The exact distribution of patterns to avoid in [S]-specified random permutation will be obtained. To achieve the goal, sampling one-by-one from an urn with [s]-specified symbols without replacement to forming random permutation. The results will cover many classical results for example the conditional runs tests and conditional scan statistics.***(iv) "Bump hunting" is vital important in genome-wide association studies between normal and disease chromosomes. In the last part of the proposal, the bump is modeled by its two components, the length and size of the bump and the joint and marginal distributions for the number and length of bumps will be derived.****B. Long term goal***(i)  The long term goal is to use FMCI technique to solve as many complex and unsolved problems, conjectures and newly arise practical problems associated with distributions of runs and patterns in applied probability and statistics, especially the continuous case. For example boundary crossing probabilities for jump processes, diffusion processes and Markov processes and matching probabilities among a set of DNA sequences allowing at most d mutations/deletions.******