Eigenvalue distributions of high-dimensional matrix processes driven by fractional Brownian motion

Pub Date : 2024-05-20 DOI:10.1142/s2010326324500096

Jian Song, Jianfeng Yao, Wangjun Yuan

引用次数: 0

Abstract

In this paper, we study high-dimensional behavior of empirical spectral distributions ${L_{N} (t), t \in [0, T]}$ for a class of $N \times N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H \in (1 / 2, 1)$ . For Wigner-type matrices, we obtain almost sure relative compactness of ${L_{N} (t), t \in [0, T]}_{N \in ℕ}$ in $C ([0, T], P (ℝ))$ following the approach in [1]; for Wishart-type matrices, we obtain tightness of ${L_{N} (t), t \in [0, T]}_{N \in ℕ}$ on $C ([0, T], P (ℝ))$ by tightness criterions provided in Appendix B. The limit of ${L_{N} (t), t \in [0, T]}$ as $N \to \infty$ is also characterized.

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分数布朗运动驱动的高维矩阵过程的特征值分布

本文研究了一类 N×N 对称/赫米特随机矩阵的经验谱分布 {LN(t),t∈[0,T]}的高维行为，这些矩阵的条目由分数布朗运动驱动的随机微分方程的解生成，赫斯特参数为 H∈(1/2,1)。对于 Wigner 型矩阵，我们按照 [1] 中的方法得到了 C([0,T],P(ℝ) 中 {LN(t),t∈[0,T]}N∈ℕ 的几乎确定的相对紧凑性；］对于 Wishart 型矩阵，我们通过附录 B 中提供的严密性判据得到 {LN(t),t∈[0,T]}N∈ℕ 在 C([0,T],P(ℝ)) 上的严密性。{LN(t),t∈[0,T]}随 N→∞ 的极限也被表征。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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