Fluctuations for matrix-valued Gaussian processes

Abstract

We consider a symmetric matrix-valued Gaussian process $Y^{(n)}=(Y^{(n)}(t);t\ge 0)$ and its empirical spectral measure process ${\mu }^{(n)}=({\mu }_t^{(n)};t\ge 0)$. Under some mild conditions on the covariance function of $Y^{(n)}$, we find an explicit expression for the limit distribution of $$Z_F^{(n)}:=\left((Z_{f_1}^{(n)}(t),\dots ,Z_{f_r}^{(n)}(t));t\ge 0\right),$$ where $F=(f_1,\dots ,f_r)$, for $r\ge 1$, with each component belonging to a large class of test functions, and $$Z_f^{(n)}(t):=n{\int }_{\mathbb{R}}f(x){\mu }_t^{(n)}(\text{d}x)-n\mathbb{E}\left[{\int }_{\mathbb{R}}f(x){\mu }_t^{(n)}(\text{d}x)\right].$$ 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 $t\ge 0$ fixed, is also given.