随机马尔可夫乘积矩阵的极限经验谱分布

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摘要：

设 {X^（k）_ij，1≤i,j≤ N；1≤k≤m}是一列独立同分布的非负实随机变量，均值为1，有限方差σ²>0,M^（k）) =(X^（k）_ij/

∑^N_j=1X^（k）_ij)_NXN(1≤k≤m)是m个马尔可夫随机矩阵。证明了当 N 趋于无穷大时，实对称矩阵σ^-2MⅡ^T_MⅡ_M的经验谱分布函数1/n∑^N_i₌₁1δλi几乎必然弱收敛于 Fuss-Catalan 分布,其中Ⅱ_M = N^M/2M^(m)…M⁽²⁾M⁽¹⁾,λ₁,λ2…,λ_N为Ⅱ^T_MⅡ_M的特征值,并通过数值模拟，得到模拟结果与理论结论一致。

Ley{X^（k）_ij，1≤i,j≤ N；1≤k≤m} be a sequence of independent and identically distributed non-negative random variables, with mean 1, finite varianceσ²>0,M^（k）) =(X^（k）_ij/∑^N_j=1X^（k）_ij)_NXN(1≤k≤m)is a sequence of Markov random matrices. In this paper, we prove that as N tends to infinity 1/n∑^N_i₌₁1δλi the empirical spectral distribution of σ^-2MⅡ^T_MⅡ_Mconverges weakly to the Fuss-Catalan distribution almost surely, where Ⅱ_M = N^M/2M^(m)…M⁽²⁾M⁽¹⁾,λ₁,λ2…,λ_N andⅡ^T_MⅡ_M are the eigenvalues of the real symmetric matrix Ⅱ^T_MⅡ_M, And the simulation results obtained through numerical simulation are consistent with the theoretical conclusion.

作者:

程慧慧，宋敏杰

Cheng Huihui,Song Minjie

机构地区：

华北水利水电大学数学与统计学院

引用本文：

程慧慧，宋敏杰。随机马尔可夫乘积矩阵的极限经验谱分布[J].学报(自然科学版),2025,53(6):52-57.(Cheng Huihui, Song Minjie.Limit of empirical spectral distribution for random Markov product matrices[J].Journal of Henan Normal University(Natural Science Edition),2025,53(6):52-57.DOI:10.16366/j.cnki.1000-2367.2024.10.14.0002.)

基金：

河南省高等学校重点科研项目

关键词：

随机马尔可夫矩阵；经验谱分布；乘积矩阵；Fuss-Catalan 分布

MMarkov random matrices ; empirical spectral distribution; product matrices; Fuss-Catalan distribution

分类号：

O211

随机马尔可夫乘积矩阵的极限经验谱分布.pdf