arXiv 1801.021112018-01-07math.PR

高斯矩阵值过程经验谱分布的收敛性

Convergence of the empirical spectral distribution of Gaussian matrix-valued processes

Arturo Jaramillo, Juan Carlos Pardo, José Luis Pérez

摘要：

对于给定的规范化高斯对称矩阵值过程$Y^{(n)}$，我们考虑其特征值过程$\{(λ_{1}^{(n)}(t),\dots, λ_{n}^{(n)}(t)); t\ge 0\}$以及相应的经验谱测度过程$μ^{(n)}=(μ_{t}^{(n)}; t\geq0)$ .在与$Y^{(n)}$相关的协方差函数的一些温和的条件下，我们证明了过程$μ^{(n)}$在紧集上一致收敛拓扑中概率收敛到确定性极限$μ$。我们证明了过程$μ$的特征在于它的柯西变换，这是一个重新标度的Burgers方程的解决方案。当$H>1/2$的论证强地利用了特征值的非碰撞性时，我们的结果推广了Rogers和Shi对自由布朗运动的结果和Pardo等人对非对易分数布朗运动的结果.我们的方法不需要后者的属性，特别是解释剩余的情况下，非交换分数布朗运动的$H< 1/2$，据我们所知，是未知的。

英文摘要：

For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(λ_{1}^{(n)}(t),\dots, λ_{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $μ^{(n)}=(μ_{t}^{(n)}; t\geq0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $μ^{(n)}$ converges in probability to a deterministic limit $μ$, in the topology of uniform convergence over compact sets. We show that the process $μ$ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers' equation. Our results extend those of Rogers and Shi for the free Brownian motion and Pardo et al. for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.

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