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

有限马尔可夫链嵌入（FMCI）技术是一种非常规的、简单、灵活和计算效率高的概率工具，用于评估运行和感兴趣模式的概率和分布。它已经成功地应用于解决健康科学、基因组分析、可靠性、质量控制、物理学、统计学、应用概率、计算机科学和离散数学等各个领域的复杂和未解决的问题。在本提案中，FMCI技术将扩展到四个非常重要的应用领域，(i)高维布朗运动的边界穿越概率（BCP）， （ii）允许最多d个突变的两个DNA序列的匹配概率，（iii）在[S]指定的随机排列中避免的模式分布，以及（iv）用于比较正常和疾病染色体之间基因表达的全基因组关联研究的凸起分布。预期结果如下：

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: