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On the Gaussian approximation of convolutions under multidimensional analogues of S. N. Bernstein’s inequality conditions. (English) Zbl 0612.60031
Let F be the distribution function of a sum of independent k-dimensional random vectors, and let $$\Phi$$ be the Gaussian distribution with zero mean and the same covariance operator as F. The main results consider the closeness of F and $$\Phi$$ ; in particular, an upper bound on the Lévy- Prohorov distance between F and $$\Phi$$ is presented, and shown to be optimal, in the case of infinitely divisible distributions and certain distributions with exponentially-decreasing tails.
The proofs are based on techniques used by V. V. Yurinskij [Teor. Veroyatn. Primen. 22, 675-688 (1977; Zbl 0399.60022)]. Similar approximation results are given for density functions and conditional distributions.
Reviewer: R.J.Tomkins

##### MSC:
 60F10 Large deviations 60E07 Infinitely divisible distributions; stable distributions 60F99 Limit theorems in probability theory
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##### References:
 [1] Ale?kevi?iene, A.K.: Multidimensional integral limit theorems for large deviations (in Russian). Teor. Veroyatn. i Primen. 28, 62-82 (1983) [2] von Bahr, B.: Multi-dimensional integral limit theorem for large deviations. Arkiv för Mat. 7, 89-99 (1967) · Zbl 0221.60015 · doi:10.1007/BF02591679 [3] Borovkov, A.A., Rogozin, B.A.: On the multidimensional central limit theorem. Teor. Veroyatn. i Primen 10, 61-69 (1965) · Zbl 0139.35206 [4] Borovkov, A.A., Sahanenko, A.I.: Estimates for convergence rate in the invariance principle for Banach spaces (in Russian). Teor. Veroyatn. i Primen. 25, 734-744 (1980) [5] Borovkov, A.A., Sahanenko, A.I.: On the rate of convergence in invariance principle. Lect. Notes Math. 1021, 59-68. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0516.60030 [6] Dudley, R.M.: Distances of probability measures and random variables. Ann. Math. Stat. 39, 1563-1572 (1968) · Zbl 0169.20602 [7] Dunnage, J.E.A.: Inequalities for the concentration functions of sums of independent random variables. Proc. London Math. Soc. 23, 489-514 (1971) · Zbl 0231.60041 · doi:10.1112/plms/s3-23.3.489 [8] Esseen, C.-G.: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math. 77, 1-125 (1945) · Zbl 0060.28705 · doi:10.1007/BF02392223 [9] Hall, P.: Bounds of the rate of convergence of moments in the central limit theorem. Ann. Probab. 10, 1004-1018 (1982) · Zbl 0516.60025 · doi:10.1214/aop/1176993721 [10] Mukhin, A.B.: Upper bounds for integrals of products of characteristic functions (in Russian). In: Limit theorems and mathematical statistics. Tashkent: Fan 1976, pp. 111-117 [11] Mukhin, A.B.: Local limit theorems for densities of sums of independent random vectors II (in Russian). Izv. Akad. Nauk Uzb. SSR Ser. Fiz-Mat. Nauk, 1, 32-35 (1984) · Zbl 0564.60018 [12] Osipov, L.V.: On the probabilities of large deviations for sums of independent random vectors (in Russian). Teor. Veroyatn. i Primen. 23, 510-526 (1978) · Zbl 0437.60005 [13] Petrov, V.V.: Sums of independent random variables. Berlin, Heidelberg, New York: Springer 1975 · Zbl 0322.60043 [14] Richter, W.: Multidimensional local limit theorems for large deviations (in Russian). Teor. Veroyatn. i Primen. 3, 107-114 (1958) [15] Sahanenko, A.I.: The rate of convergence in the invariance principle for non-identically distributed random variables with exponential moments (in Russian). In: Limit theorems for sums of random variables, Trudy Instituta matematiki SO AN SSSR, v. 3, pp. 4-49. Novosibirsk: Nauka 1984 [16] Saulis, L.: On large deviations of random vectors for some classes of sets I, II (in Russian). Lit. mat. sb. 23, 3, 142-154; 44, 50-57 (1983) [17] Statulevi?ius, V.A.: On large deviations. Z. Wahrscheinlichkeitstheor. Verw. Geb. 6, 133-144 (1966) · Zbl 0158.36207 · doi:10.1007/BF00537136 [18] Yurinski i, V.V.: Exponential inequalities for sums of random vectors. J. Multivariate Anal. 46, 473-499 (1976) · Zbl 0346.60001 · doi:10.1016/0047-259X(76)90001-4 [19] Yurinski i, V.V.: On approximation of convolutions by normal laws (in Russian). Teor. Veroyatn. i Primen 22, 675-688 (1977) [20] Za itsev, A.Yu., Arak, T.V.: On the rate of convergence in the second uniform limit theorem of Kolmogorov (in Russian). Teor. Veroyatn. i Primen. 28, 333-353 (1983) [21] Za itsev, A.Yu.: Some remarks on the approximation of sums of independent terms (in Russian) Zapiski nauchnyh seminarov LOMI 136, 48-57 (1984) [22] Za itsev, A.Yu.: On the approximation by Gaussian distributions with satisfaction of multidimensional analogues of Bernstein’s conditions (in Russian). Doklady Akademii nauk SSSR 276, 1046-1048 (1984) [23] Za itsev, A.Yu: On the Gaussian approximation of convolutions under multidimensional analogues of Bernstein’s inequality conditions (in Russian). Preprint LOMI P-9-84. Leningrad 1984 [24] Za itsev, A.Yu.: On the approximation of convolutions of multidimensional distributions (in Russian). Zapiski nauchnyh seminarov LOMI 142, 68-80 (1985) [25] Zolotarev, V.M.: Metric distances in spaces of random variables and of their distributions (in Russian). Mat. Sb. (N.S.) 101, 416-454 (1976) · Zbl 0367.60003
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