非稳态马尔可夫链服从Benford定律
Finite-state Markov Chains obey Benford's Law
摘要：
一个实数序列（x_n）是Benford的，如果有效数，即（x_n）的浮点表示中的分数部分是几何分布的。类似地，具有概率转移阵P和极限矩阵P* 的离散时间不可约非周期有限状态马尔可夫链是Benford，如果矩阵序列（P^n - P*）和（P^{n+1}-P^n）的每个分量都是Benford或最终为零。利用最近的工具，建立本福德行为的牛顿法和有限维线性映射，通过均匀分布模1和Perron-Frobenius的经典理论，本文推导出一个简单的充分条件（非共振），保证P，或与之相关的马尔可夫链，是本福德。这个结果反过来又用来表明，几乎所有的马尔可夫链都是本福德，在这个意义上，如果转移概率是独立和连续选择的，那么所得到的马尔可夫链是本福德概率为1。具体的例子说明了出现的各种情况下，和理论的补充与几个模拟和潜在的应用。
英文摘要：
A sequence of real numbers (x_n) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x_n) are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (P^n - P*) and (P^{n+1}-P^n) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton's method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition (nonresonant) guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probabilities are chosen independently and continuously, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with several simulations and potential applications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
