Abstract
Two–sided bounds are constructed for a probability density function of a weighted sum of chi-square variables. Both cases of central and non-central chi-square variables are considered. The upper and lower bounds have the same dependence on the parameters of the sum and differ only in absolute constants. The estimates obtained will be useful, in particular, when comparing two Gaussian random elements in a Hilbert space and in multidimensional central limit theorems, including the infinite-dimensional case.
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Acknowledgements
Theorem 1 has been obtained under support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No 075-15-2019-1621. Theorem 2 was proved within the framework of the HSE University Basic Research Program. Research of S. Bobkov was supported by the NSF grant DMS-1855575.
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Bobkov, S.G., Naumov, A.A., Ulyanov, V.V. (2021). Two–Sided Bounds for PDF’s Maximum of a Sum of Weighted Chi-square Variables. In: Shiryaev, A.N., Samouylov, K.E., Kozyrev, D.V. (eds) Recent Developments in Stochastic Methods and Applications. ICSM-5 2020. Springer Proceedings in Mathematics & Statistics, vol 371. Springer, Cham. https://doi.org/10.1007/978-3-030-83266-7_13
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