Geodesic
On a~hitting probability of Gaussian random vector into a~small ball in a~Hilbert space
Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 75-93

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Let $\xi$ be a Gaussian random vector taking its value in a Hilbert space $H$. Denote by $\theta(a,z)$ the ball in $H$ with center $a$ and radius $z$. Let $I(a,z)=\Prob\{\xi\in\theta(a,z)\}$, $z\to0$. We give some asymptotic formulas for $I(a,z)$ valid when $z\to0$. 
@article{ZNSL_1979_85_a5,
     author = {I. A. Ibragimov},
     title = {On a~hitting probability of {Gaussian} random vector into a~small ball in {a~Hilbert} space},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {75--93},
     publisher = {mathdoc},
     volume = {85},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a5/}
}
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I. A. Ibragimov. On a~hitting probability of Gaussian random vector into a~small ball in a~Hilbert space. Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 75-93. http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a5/