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、灵活和计算高效的概率工具，用于评估游程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领域的复杂和悬而未决的问题。在这个建议中，FMCI技术将被扩展到四个非常重要的应用领域，(I)高维布朗运动的边界跨越概率，(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点分布。以下是预期结果：*A.短期预期结果(未来五年)*(I)建议的第一部分将建立一种分析和有效的数值方法来逼近d维布朗运动的非线性凸边界的越界概率。所得结果将推广到Ornstein-Uhlenbeck过程和布朗桥过程等相关的随机过程。*(Ii)在统计量Ln(D)中，提出了最多允许d个突变/插入的两个DNA序列的最长匹配长度作为相似性的度量。导出了Ln(D)的精确分布，并证明了Ln(D)的分布可以用扫描统计量的分布来表示。*(Iii)得到了在[S]指定的随机排列中要避免的图案的精确分布。为了达到这一目的，从带有[S]指定符号的骨灰盒中逐一采样，而不进行替换以形成随机排列。结果将涵盖许多经典结果，例如条件游程检验和条件扫描统计。*(Iv)在正常染色体和疾病染色体之间的全基因组关联研究中，“Bump Hunting”是至关重要的。在提案的最后部分，凸起由它的两个组成部分来模拟，凸起的长度和大小以及凸起数量和长度的联合和边缘分布将被推导出来。*b.长期目标*(I)长期目标是使用FMCI技术来解决应用概率和统计学中与游程分布和模式有关的尽可能多的复杂和未解决的问题、猜想和新出现的实际问题，特别是连续情况。例如，跳跃过程、扩散过程和马尔可夫过程的边界跨越概率以及允许最多d个突变/缺失的一组DNA序列之间的匹配概率。

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.******