Abstract
An improvement of a result of Le Cam on the rate of approximation of distributions of sums of independent random variables by accompanying compound Poisson laws is proved. Bibliography: 14 titles.
Similar content being viewed by others
References
T. V. Arak, “On the rate of convergence in Kolmogorov’s uniform limit theorem,” Teor. Veroyatn. Primen, 26, 225–245 (1981).
T. V. Arak and A. Yu. Zaitsev, Uniform Limit Theorems for Sums of Independent Random Variables [in Russian], Trudy Steklov Mat. Inst., 174 (1986).
A. D. Barbour and O. Chryssaphinou, “Compound Poisson approximation: a user’s guide,” Ann. Appl. Probab., 11, No. 3, 964–1002 (2001).
V. Čekanavičius, “On the approximation by convolutions of the generalized Poisson measure and the Gaussian distribution. I, II,” Lithuanian Math. J., 32, No. 3, 265–274; No. 4, 355–361 (1992).
W. Hengartner and R. Theodorescu, Concentration Functions, Academic Press, New York—London (1973).
I. A. Ibragimov and E. L. Presman, “The rate of convergence of the distributions of sums of independent random variables to accompanying laws,” Teor. Veroyatn. Primen., 18, 753–766 (1973).
M. Katz, “Note on the Berry—Esseen theorem,” Ann. Math. Statist., 34, No. 4, 1107–1108 (1963).
A. N. Kolmogorov, “Approximation of distributions of sums of independent terms by infinitely divisible distributions,” Trudy Moskov. Mat. Obsc., 12, 437–451 (1963).
L. Le Cam, “On the distribution of sums of independent random variables,” in: Bernoulli (1713), Bayes (1963), Laplace (1813). Anniversary Volume, Springer-Verlag, New York (1965), pp. 179–202.
L. Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer-Verlag, New York (1986).
V. V. Petrov, “A bound for the deviation of the distribution of a sum of independent random variables from the normal law,” Dokl. Akad. Nauk SSSR, 160, No. 5, 1013–1015 (1965).
A. Yu. Zaitsev, “On the uniform approximation of functions of the distribution of sums of independent random variables,” Teor. Veroyatn. Primen., 32, No. 1, 45–52 (1987).
A. Yu. Zaitsev, “Approximation of convolutions of probability distributions by in.nitely divisible laws under weakened moment constraints,” Zap. Nauchn. Semin. POMI, 194, 79–90 (1992).
A. Yu. Zaitsev, “On the approximation of a sample by a Poisson point process,” Zap. Nauchn. Semin. POMI, 298, 111–125 (2003).
Author information
Authors and Affiliations
Additional information
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 44–53.
Rights and permissions
About this article
Cite this article
Götze, F., Zaitsev, A.Y. Approximation of convolutions by accompanying laws without centering. J Math Sci 137, 4510–4515 (2006). https://doi.org/10.1007/s10958-006-0243-2
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0243-2