One considers the Kawata and the Kunisawa concentration functions. For these concentration functions one proves the analogues of Kesten's inequality.
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T. Kawata, “The function of mean concentration of a chance variable,” Duke Math. J.,8, No. 4, 666–677 (1941).
K. Kunisawa, “On an analytical method in the theory of independent random variables,” Ann. Inst. Statist. Math.,1, 1–77 (1949).
S. M. Anan'evskii, “On concentration functions,” Vestn. Leningr. Univ., No. 13, 7–13 (1978).
H. Kesten, “A sharper form of the Doeblin-Levy-Kolmogorov-Rogozin inequality for concentration functions,” Math. Scand.,25, 133–144 (1969).
L. P. Postnikova and A. A. Yudin, “An analytic method for the estimates of the concentration functions,” Tr. Mat. Inst. Akad. Nauk SSSR,143, 143–151 (1977).
V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag, New York (1975).
G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press, London (1952).
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 15–21, 1980.
In conclusion, the author expresses his gratitude to V. V. Petrov for suggesting this topic and for his constant interest in the paper.
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Anan'evskii, S.M. Analogues of the Kesten inequality for concentration functions and their refinements. J Math Sci 24, 490–494 (1984). https://doi.org/10.1007/BF01702324
- Concentration Function