论文标题
具有马尔可夫依赖性块的密集随机矩阵的奇异值分布
Singular value distribution of dense random matrices with block Markovian dependence
论文作者
论文摘要
马尔可夫链是一个马尔可夫链，其状态空间可以分为有限数量的群集，因此过渡概率仅取决于簇。因此，马尔可夫连锁店是马尔可夫连锁社区的模型。本文建立了限制定律，以实现与块马尔可夫链的样本路径相关的经验过渡矩阵和经验频率矩阵的奇异价值分布，只要样本路径的长度为$θ（n^2）$，而状态空间的大小则是$θ（n^2）$。 证明方法分为两部分。首先，我们介绍了一类对称随机矩阵，其依赖条目称为近似不相关的随机矩阵，具有方差曲线。我们通过瞬间方法建立了他们的限制特征值分布。其次，我们开发了一个耦合参数，以表明这种通用结果适用于与Block Markov链相关的奇异值分布。
A block Markov chain is a Markov chain whose state space can be partitioned into a finite number of clusters such that the transition probabilities only depend on the clusters. Block Markov chains thus serve as a model for Markov chains with communities. This paper establishes limiting laws for the singular value distributions of the empirical transition matrix and empirical frequency matrix associated to a sample path of the block Markov chain whenever the length of the sample path is $Θ(n^2)$ with $n$ the size of the state space. The proof approach is split into two parts. First, we introduce a class of symmetric random matrices with dependent entries called approximately uncorrelated random matrices with variance profile. We establish their limiting eigenvalue distributions by means of the moment method. Second, we develop a coupling argument to show that this general-purpose result applies to the singular value distributions associated with the block Markov chain.
