Comparisons between tail probabilities of sums of independent symmetric random variables
Annales de l'I.H.P. Probabilités et statistiques, Tome 33 (1997) no. 5, pp. 651-671. 
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     author = {Pruss, Alexander R.},
     title = {Comparisons between tail probabilities of sums of independent symmetric random variables},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {651--671},
     publisher = {Gauthier-Villars},
     volume = {33},
     number = {5},
     year = {1997},
     mrnumber = {1473569},
     zbl = {0893.60009},
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     url = {http://archive.numdam.org/item/AIHPB_1997__33_5_651_0/}
}
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Pruss, Alexander R. Comparisons between tail probabilities of sums of independent symmetric random variables. Annales de l'I.H.P. Probabilités et statistiques, Tome 33 (1997) no. 5, pp. 651-671. http://archive.numdam.org/item/AIHPB_1997__33_5_651_0/

[1] A. Bikelis [= BIKYALIS], On estimates of the remainder term in the central limit theorem, Litovskiĭ Mat. Sb., Vol. 6, 1966, p. 323-346 (Russian). | MR | Zbl

[2] D.L. Burkholder, Explorations in martingale theory and its applications, Ecole d'Eté de Probabilités de Saint-Flour XIX - 1989 (D. L. Burkholder, E. Padoux and A. Sznitman, eds.), Lecture Notes in Mathematics, Vol. 1464, Springer-Verlag, New York, 1991, p. 1-66. | MR | Zbl

[3] R. Chen, A remark on the tail probability of a distribution, J. Multivariate Analysis, Vol. 8, 1978, p. 328-333. | MR | Zbl

[4] P. Erdös, On a theorem of Hsu and Robbins, Ann. Math. Statist., Vol. 20, 1949, p. 286-291. | MR | Zbl

[5] P. Erdös, Remark on my paper "On a theorem of Hsu and Robbins", Ann. Math. Statist., Vol. 21, 1950, p. 138. | MR | Zbl

[6] B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, revised, Addison-Wesley, Reading, Massachusetts, 1968. | MR

[7] A. Gut, Complete convergence, Asymptotic Statistics: Proceedings of the Fifth Symposium held at Charles University, Prague, September 4-9, 1993 (P. Mandl and M. Hušková, eds.), Contrib. Statist., Physica-Verlag, Heidelberg. | MR

[8] A. Gut, Complete convergence for arrays, Periodica Math. Hungarica, Vol. 25, 1992, p. 51-75. | MR | Zbl

[9] C.C. Heyde, A supplement to the strong law of large numbers, J. Appl. Prob., Vol. 12, 1975, p. 173-175. | MR | Zbl

[10] P.L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A., Vol. 33, 1947, p. 25-31. | MR | Zbl

[11] C.S. Kahane, Evaluating Lebesgue integrals as limits of Riemann sums, Math. Japonica, Vol. 38, 1993, p. 1073-1076. | MR | Zbl

[12] O.I. Klesov, Convergence of series of probabilities of large deviations of sums of independent identically distributed random variables, Ukraïnskiĭ Mat. Zhurn., Vol. 45, 1993, p. 770-784. | MR | Zbl

[13] S. Kwapień and W.A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhauser, Boston, 1992. | MR | Zbl

[14] D. Li, M.B. Rao, T. Jiang and X. Wang, Convergence and almost sure convergence of weighted sums of random variables, J. Theoret. Probab., Vol. 8, 1995, p. 49-76. | MR | Zbl

[15] M. Loève, Probability Theory, 3rd edition, Van Nostrand, New York, 1963. | MR | Zbl

[16] S.J. Montgomery-Smith, Comparison of sums of identically distributed random vectors, Probab. Math. Statist., Vol. 14, 1993, p. 281-285. | MR | Zbl

[17] A.R. Pruss, On Spătaru's extension of the Hsu-Robbins-Erdös law of large numbers, J. Math. Anal. Appl., Vol. 199, 1996, p. 558-578. | MR | Zbl

[18] A.R. Pruss, Randomly sampled Riemann sums and complete convergence in the law of large numbers for a case without identical distribution, Proc. Amer. Math. Soc., Vol. 124, 1996, p. 919-929. | MR | Zbl

[19] A.R. Pruss, A two-sided estimate in the Hsu-Robbins-Erdös law of large numbers, Stochastic Processes Appl. (to appear). | MR | Zbl

[20] A.R. Pruss, Remarks on summability of series formed from deviation probabilities of sums of independent identically distributed random variables, Ukraïnskiĭ Mat. Zhurn., Vol. 48, 1996, p. 569-572. | MR | Zbl

[21] A.R. Pruss, A bounded N-tuplewise independent and identically distributed counterexample to the CLT, Preprint, 1997.

[22] D. Szynal, On complete convergence for some classes of dependent random variables, Annales Univ. Mariae Curie-Sklodowska, (Sectio A), Vol. 47, 1993, p. 145-150. | MR | Zbl

[23] H. Thorisson, Coupling methods in probability theory, Scand. J. Statist., Vol. 22, 1995, p. 159-182. | MR | Zbl

[24] W.A. Woyczyński, Tail probabilities of sums of random vectors in Banach spaces, and related norms, Measure Theory Oberwolfach 1979 (D. Kolzow, ed.), Lecture Notes in Mathematics, Vol. 794, Springer-Verlag, New York, 1980, p. 455-469. | MR | Zbl

[25] W.A. Woyczyński, On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related rates of convergence, Probab. Math. Statist., Vol. 1, 1980, p. 117-131. | MR | Zbl