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28 March 2024
 
  » arxiv » 2001.3718

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Fluctuations for matrix-valued Gaussian processes
Mario Diaz ; Arturo Jaramillo ; Juan Carlos Pardo ;
Date 11 Jan 2020
AbstractWe consider a symmetric matrix-valued Gaussian process $Y^{(n)}=(Y^{(n)}(t);tge0)$ and its empirical spectral measure process $mu^{(n)}=(mu_{t}^{(n)};tge0)$. Under some mild conditions on the covariance function of $Y^{(n)}$, we find an explicit expression for the limit distribution of $$Z_F^{(n)} := left( ig(Z_{f_1}^{(n)}(t),ldots,Z_{f_r}^{(n)}(t)ig) ; tge0 ight),$$ where $F=(f_1,dots, f_r)$, for $rge 1$, with each component belonging to a large class of test functions, and $$ Z_{f}^{(n)}(t) := nint_{mathbb{R}}f(x)mu_{t}^{(n)}( ext{d} x)-nmathbb{E}left[int_{mathbb{R}}f(x)mu_{t}^{(n)}( ext{d} x) ight].$$ 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 $tgeq0$ fixed, is also given.
Source arXiv, 2001.3718
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