Fluctuations for matrix-valued Gaussian processes

Abstract

We consider a symmetric matrix-valued Gaussian process ${\mathit{Y}}^{(\mathit{n})}=({\mathit{Y}}^{(\mathit{n})}(\mathit{t});\mathit{t}\ge 0)$ and its empirical spectral measure process ${\mathit{\mu }}^{(\mathit{n})}=({\mathit{\mu }}_{\mathit{t}}^{(\mathit{n})};\mathit{t}\ge 0)$. Under some mild conditions on the covariance function of ${\mathit{Y}}^{(\mathit{n})}$, we find an explicit expression for the limit distribution of

$${\mathit{Z}}_{\mathit{F}}^{(\mathit{n})}:=\left(\right({\mathit{Z}}_{{\mathit{f}}_1}^{(\mathit{n})}(\mathit{t}),\dots ,{\mathit{Z}}_{{\mathit{f}}_{\mathit{r}}}^{(\mathit{n})}(\mathit{t}));\mathit{t}\ge 0),$$

where $\mathit{F}=({\mathit{f}}_1,\dots ,{\mathit{f}}_{\mathit{r}})$, for $\mathit{r}\ge 1$, with each component belonging to a large class of test functions, and

$${\mathit{Z}}_{\mathit{f}}^{(\mathit{n})}(\mathit{t}):=\mathit{n}{\int }_{\mathbb{R}}\mathit{f}(\mathit{x}){\mathit{\mu }}_{\mathit{t}}^{(\mathit{n})}(\mathrm{d}\mathit{x})-\mathit{n}\mathbb{E}[{\int }_{\mathbb{R}}\mathit{f}(\mathit{x}){\mathit{\mu }}_{\mathit{t}}^{(\mathit{n})}(\mathrm{d}\mathit{x})].$$

More precisely, we establish the stable convergence of ${\mathit{Z}}_{\mathit{F}}^{(\mathit{n})}$ and determine its limiting distribution. An upper bound for the total variation distance of the law of ${\mathit{Z}}_{\mathit{f}}^{(\mathit{n})}(\mathit{t})$ to its limiting distribution, for a test function f and $\mathit{t}\ge 0$ fixed, is also given.

Nous considérons un processus gaussien symétrique à valeurs matricielles ${\mathit{Y}}^{(\mathit{n})}=({\mathit{Y}}^{(\mathit{n})}(\mathit{t});\mathit{t}\ge 0)$ et son processus des mesures spectrales empiriques ${\mathit{\mu }}^{(\mathit{n})}=({\mathit{\mu }}_{\mathit{t}}^{(\mathit{n})};\mathit{t}\ge 0)$. Sous des conditions assez faibles sur la fonction de covariance de ${\mathit{Y}}^{(\mathit{n})}$ nous trouvons une expression explicite pour la loi limite de

$${\mathit{Z}}_{\mathit{F}}^{(\mathit{n})}:=\left(\right({\mathit{Z}}_{{\mathit{f}}_1}^{(\mathit{n})}(\mathit{t}),\dots ,{\mathit{Z}}_{{\mathit{f}}_{\mathit{r}}}^{(\mathit{n})}(\mathit{t}));\mathit{t}\ge 0),$$

où $\mathit{F}=({\mathit{f}}_1,\dots ,{\mathit{f}}_{\mathit{r}})$, pour $\mathit{r}\ge 1$, avec chaque composant appartenant à une grande classe des fonctions test, et

$${\mathit{Z}}_{\mathit{f}}^{(\mathit{n})}(\mathit{t}):=\mathit{n}{\int }_{\mathbb{R}}\mathit{f}(\mathit{x}){\mathit{\mu }}_{\mathit{t}}^{(\mathit{n})}(\mathrm{d}\mathit{x})-\mathit{n}\mathbb{E}[{\int }_{\mathbb{R}}\mathit{f}(\mathit{x}){\mathit{\mu }}_{\mathit{t}}^{(\mathit{n})}(\mathrm{d}\mathit{x})].$$

Plus précism´ent, nous établissons la convergence stable de ${\mathit{Z}}_{\mathit{F}}^{(\mathit{n})}$ et nous déterminons sa loi limite. Nous donnons également une borne supérieure pour la distance en variation totale entre la loi de ${\mathit{Z}}_{\mathit{f}}^{(\mathit{n})}(\mathit{t})$ et sa loi limite, pour une fonction test f et $\mathit{t}\ge 0$ fixés.