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Fluctuations for matrix-valued Gaussian processes | Mario Diaz
; Arturo Jaramillo
; Juan Carlos Pardo
; | Date: |
11 Jan 2020 | Abstract: | We 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 | Services: | Forum | Review | PDF | Favorites |
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