Geodesic
On distribution of quadratic forms in Gaussian random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 2, pp. 301-312

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Two-sided bounds are constructed for a density function $p(u,a)$ of a random variable $|Y-a|^2 $, where $Y$ is a Gaussian random element in a Hilbert space with zero mean. The estimates are sharp in the sense that starting from large enough $u$ the ratio of upper bound to lower bound equals 8 and does not depend on any parameters of a distribution of $|Y-a|^2$. The estimates imply two-sided bounds for probabilities $\mathbf{P}(|Y-a|>r)$
Keywords: Gaussian measure, tail behavior, distribution of quadratic forms.
Mots-clés : noncentral $\chi^2$-distribution 
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     author = {G. Christoph and Yu. V. Prokhorov and V. V. Ulyanov},
     title = {On distribution of quadratic forms in {Gaussian} random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {301--312},
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     volume = {40},
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     year = {1995},
     language = {ru},
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G. Christoph; Yu. V. Prokhorov; V. V. Ulyanov. On distribution of quadratic forms in Gaussian random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 2, pp. 301-312. http://geodesic.mathdoc.fr/item/TVP_1995_40_2_a4/