Abstract
We prove two theorems on upper and lower bounds for probabilities in the multidimensional case. We generalize and improve the Prokhorov multidimensional analog of the Chebyshev inequality and establish a multidimensional analog of the generalized Kolmogorov probability estimate.
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Yu. V. Prokhorov, “Multidimensional distributions: inequalities and limit theorems,” in: VINITI Series in Probability Theory, Mathematical Statistics, and Theoretical Cybernetics [in Russian], Vol. 10, VINITI, Moscow (1972), pp. 5–24.
N. V. Sokolov, “Extension of possibilities of Chebyshev-type inequalities and Kolmogorov estimates, ” Dokl. Ros. Akad. Nauk, 384, No. 3, 308–311 (2002).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 573–576, April, 2006.
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Sokolov, N.V. Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality. Ukr Math J 58, 645–650 (2006). https://doi.org/10.1007/s11253-006-0090-9
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DOI: https://doi.org/10.1007/s11253-006-0090-9