Estimates for the Rapid Decay of Concentration Functions of n-Fold Convolutions

Journal of Theoretical Probability

Volumn 11, Issue 3, 1998, Pages 715-731

  Gotze F ^a   Zaitsev, A Yu ^b  

^a BIELEFELD UNIVERSITY   (Germany)

^b STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

Author keywords

Concentration functions; n fold convolutions; Rates of decay; Sums of i.i.d. random variables

Indexed keywords

EID: 0032394579     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022654631571     Document Type: Article

References (20)

1. Arak, T. V. (1981). On the convergence rate in Kolmogorov's uniform limit theorem. I. Theor. Prob. Appl. 26, 219-239.
2. Arak, T. V., and Zaitsev, A. Yu. (1988). Uniform limit theorems for sums of independent random variables. Proc. Steklov Inst. Math. 174, 1-222.
3. Esseen, C.-G. (1968). On the concentration function of a sum of independent random variables. Z. Wahrsch. verw. Gebiete 9, 290-308.
4. Feller, W. (1967). On regular variation and local limit theorems. In Proc. of the Fifth Berkeley Symp. on Math. Stat. Prob. Vol. 2, Part 1, University of California Press, Berkeley, pp. 373-388.
5. Griffin, P. S. (1983). Probability estimates for the small deviations of d-dimensional random walk. Ann. Prob. 11, 939-952.
6. Griffin, P. S., Jain, N. C., and Pruitt, W. E. (1984). Approximate local limit theorems outside domains of attraction. Ann. Prob. 12, 45-63.
7. Hall, P. (1983). Order of magnitude of the concentration function. Proc. Amer. Math. Soc. 89, 141-144.
8. Hengartner, W., and Theodorescu, R. (1973). Concentration Functions, Academic Press, New York.
9. Morozova, L. N. (1977). Some bounds for the concentration functions of a sum of independent and identically distributed random variables. In Limit Theorems for Random Processes, S. H. Sirazhdinov, ed. Fan, Tashkent, pp. 85-91. (Russian)
10. Mukhin, A. B. (1972; 1973; 1976). On the concentration of distributions of sums of independent random variables, I; II; III. Izvestiya AN UzSSR, Ser. Fiz.-Matem. Nauk 2, 17-19; 6, 18-23; 1, 15-19. (Russian)
11. Mukhin, A. B. (1972; 1973; 1976). On the concentration of distributions of sums of independent random variables, I; II; III. Izvestiya AN UzSSR, Ser. Fiz.-Matem. Nauk 2, 17-19; 6, 18-23; 1, 15-19. (Russian)
12. Mukhin, A. B. (1972; 1973; 1976). On the concentration of distributions of sums of independent random variables, I; II; III. Izvestiya AN UzSSR, Ser. Fiz.-Matem. Nauk 2, 17-19; 6, 18-23; 1, 15-19. (Russian)
13. Mukhin, A. B. (1976). An estimate of the rapid decay of concentration functions of sums of independent random variables In Limit Theorems and Mathematical Statistics, S. H. Sirazhdinov, ed. Fan, Tashkent, pp. 117-121. (Russian)
14. Mukhin, A. B. (1989). Local probabilities for sums of independent random variables. Theor. Prob. Appl. 34, 616-624.
15. Petrov, V. V. (1976). Sums of Independent Random Variables, Springer, Berlin.
16. Rogozin, B. A. (1961). An estimate for concentration functions. Theor. Prob. Appl. 6, 94-97.
17. Studnev, Yu. P. (1965). A remark on the Katz-Petrov theorem. Theor. Prob. Appl. 10, 682-684.
18. Suchkov, A. P., and Ushakov, N. G. (1989). The rapid decay of concentration functions of sums of independent random variables. Theor. Prob. Appl. 34, 545-548.
19. Zaitsev, A. Yu. (1987). On the uniform approximation of distributions of sums of independent random variables. Theor. Prob. Appl. 32, 40-47.
20. Zaitsev, A. Yu. (1992). Approximation of convolutions of probability distributions by infinitely divisible laws under weakened moment restrictions. Zapiski Nauchnykh Seminarov LOMI 194, 79-90 (Russian); English transl, in J. Math. Sci.