The aim of the present work is evaluation of absolute constants in the Arak inequalities for the concentration functions of convolutions of probability distributions. This result allows us to calculate the constant in the inequality for the uniform distance between n and (n + 1)-fold convolutions of one-dimensional symmetric probability distributions with a characteristic function separated from −1, as well as a number of other estimates, in particular, the accuracy of approximation of samples of rare events by the Poisson point process.
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T. V. Arak and A. Yu. Zaitsev, “Uniform limit theorems for sums of independent random variables,” Trudy MIAN, 174, 3–214 (1986).
B. Roos, “On Hipp’s compound Poisson approximations via concentration functions,” Bernoulli, 11, 533–557 (2005).
Ya. S. Golikova, “On improvement of the estimate for the distance between distributions of sequential sums of independent random variables,” Zap. Nauchn. Semin. POMI, 474, 118–123 (2018); English transl. J. Math. Sci., 251, No. 1, 74–77 (2020).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 486, 2019, pp. 86–97.
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Golikova, Y.S. On Calculation of Constants in the Arak Inequalities for Concentration Functions of Convolutions of Probability Distributions. J Math Sci 258, 793–801 (2021). https://doi.org/10.1007/s10958-021-05581-2
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DOI: https://doi.org/10.1007/s10958-021-05581-2