arXiv 2001.037182020-01math.PR

矩阵值高斯过程的涨落

Fluctuations for matrix-valued Gaussian processes

Mario Diaz, Arturo Jaramillo, Juan Carlos Pardo

摘要：

考虑对称矩阵值高斯过程$Y^{(n)}=(Y^{(n)}(t);t\ge0)$及其经验谱测度过程$μ^{(n)}=(μ_{t}^{(n)};t\ge0)$ .在对$Y^{(n)}$的协方差函数的适当条件下，我们得到了$$Z_F^{(n)} := \left( \big(Z_{f_1}^{(n)}(t),\ldots,Z_{f_r}^{(n)}(t)\big) ; t\ge0\right),$$的极限分布的显式表达式，其中$F=(f_1,\dots, f_r)$，对于$r\ge 1$，每个分量属于一个大的测试函数类，而$$ Z_{f}^{(n)}(t) := n\int_{\mathbb{R}}f(x)μ_{t}^{(n)}(\text{d} x)-n\mathbb{E}\left[\int_{\mathbb{R}}f(x)μ_{t}^{(n)}(\text{d} x)\right].$$更精确地说，我们建立了$Z_F^{(n)}$的稳定收敛性，并确定了它的极限分布.当测试函数$f$和$t\geq0$固定时，给出了$Z_{f}^{(n)}(t)$定律到其极限分布的总变化距离的上界.

英文摘要：

We consider a symmetric matrix-valued Gaussian process $Y^{(n)}=(Y^{(n)}(t);t\ge0)$ and its empirical spectral measure process $μ^{(n)}=(μ_{t}^{(n)};t\ge0)$. Under some mild conditions on the covariance function of $Y^{(n)}$, we find an explicit expression for the limit distribution of $$Z_F^{(n)} := \left( \big(Z_{f_1}^{(n)}(t),\ldots,Z_{f_r}^{(n)}(t)\big) ; t\ge0\right),$$ where $F=(f_1,\dots, f_r)$, for $r\ge 1$, with each component belonging to a large class of test functions, and $$ Z_{f}^{(n)}(t) := n\int_{\mathbb{R}}f(x)μ_{t}^{(n)}(\text{d} x)-n\mathbb{E}\left[\int_{\mathbb{R}}f(x)μ_{t}^{(n)}(\text{d} x)\right].$$ More precisely, we establish the stable convergence of $Z_F^{(n)}$ and determine its limiting distribution. An upper bound for the total variation distance of the law of $Z_{f}^{(n)}(t)$ to its limiting distribution, for a test function $f$ and $t\geq0$ fixed, is also given.

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