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）技术是一种非传统的，简单，灵活和计算效率高的概率工具，以评估的概率和分布的运行和感兴趣的模式。它已成功应用于解决各种领域的复杂和未解决的问题，如健康科学，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。本文将FMCI技术推广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）最多允许d个突变的两个DNA序列的匹配概率，（iii）在[S]-指定的随机排列中要避免的模式的分布和（iv）基因组-广泛的关联研究，用于比较正常和疾病染色体之间的基因表达。预期成果如下：

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: