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A metric approach to investigation of the stability of Pólya theorem on characterization of the normal distribution

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Literature cited

  1. A. M. Kagan, Yu. V. Linnik, and S. R. Rao, Characterization Problems of Mathematical Statistics [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  2. Yu. R. Gabovich, “On stability of some characteristic properties of a normal population,” Teor. Veroyatn. Primen.,19, No. 2, 381–389 (1974).

    Google Scholar 

  3. Yu. R. Gabovich, “Stability of the characterization of the normal distribution in Skitovich-Darmois theorem,” J. Sov. Math.,16, No. 5 (1981).

  4. V. M. Zolotarev, “The stability effect of the characterization of a distribution,” J. Sov. Math.,16, No. 5 (1981).

  5. V. M. Zolotarev, “On continuity of stochastic sequences generated by recursive procedures,” Teor. Veroyatn. Primen.,20, No. 4, 834–847 (1975).

    Google Scholar 

  6. V. M. Zolotarev, “Metric distances in spaces of random variables and of their distributions,” Mat. Sb. (N.S.),101, No. 3, 416–454 (1976).

    Google Scholar 

  7. V. M. Zolotarev, “On pseudomoments,” Teor. Veroyatn. Primen.,23, No. 2, 284–294 (1978).

    Google Scholar 

  8. V. M. Zolotarev, “Ideal metrics in the problem of approximating the distributions of sums of independent random variables,” Teor. Veroyatn. Primen.,22, No. 3, 449–469 (1977).

    Google Scholar 

  9. V. I. Rotar', “Nonclassical estimates of the rate of convergence in the multidimensional central limit theorem,” Teor. Veroyatn. Primen.,22, No. 4, 774–790 (1977);23, No. 1, 55–66 (1978).

    Google Scholar 

  10. V. V. Ul'yanov, “On the stability of estimates of the rate of convergence in the central limit theorem,” Teor. Veroyatn. Primen.,23, No. 3, 684–687 (1978).

    Google Scholar 

  11. V. V. Yurinskii, “A smoothing inequality for estimates of Levy-Prokhorov distance,” Teor. Veroyatn. Primen.,20, No. 1, 3–12 (1975).

    Google Scholar 

  12. A. Mitalauskas, “Asymptotic expansion for independent random variables in the case of a stable limit distribution,” Litov. Mat. Sb.,3, No. 1 (1963).

  13. V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag (1975).

  14. M. Loeve, Probability Theory, 2nd ed., Van Nostrand, Princeton, New Jersey (1960).

    Google Scholar 

  15. B. I. Rotar', “A nonuniform estimate of the rate of convergence in the central limit theorem,” Teor. Veroyatn. Primen.,25, No. 4, 647–665 (1980).

    Google Scholar 

  16. A. A. Zinger and R. V. Yanushkyavichyus, “On a characterization of the normal law and its stability,” Teor. Veroyatn. Primen.,25, No. 2, 425–427 (1980).

    Google Scholar 

  17. R. V. Yanushkyavichyus, “Estimates of stability of the characterization of the normal law in Pólya theorem,” J. Sov. Math.,17, No. 6 (1981).

  18. I. S. Shiganov, Candidate's Dissertation, Moscow (1979).

  19. I. S. Shiganov, “On multidimensional analogues of Gram-Charlier and Edgeworth-Cramer expansions,” in: Probl. Ustoichiv. Stokhast. Modelei. Trudy Semin., VNIISI, Moscow (1980), pp. 116–121.

    Google Scholar 

  20. V. M. Zolotarev, “On a lemma of Yu. R. Gabovich,” Teor. Veroyatn. Primen.,25, No. 2, 423–424 (1980).

    Google Scholar 

  21. Yu. R. Gabovich, “Stability of some characterizations of the normal distribution,” Candidate's Dissertation, Dolgoprudnyi (1974).

  22. A. Bikyalis, “Estimates of the remainder in the central limit theorem,” Litov. Mat. Sb.,6, No. 3, 323–346 (1976).

    Google Scholar 

  23. L. D. Meshalkin, “On the robustness of some characterization of the normal distribution,” Ann. Math. Stat.,39, No. 5, 2747–2750 (1968).

    Google Scholar 

  24. V. M. Zolotarev, “General problems of the stability of mathematical models,” Bull. Int. Stat. Inst.,47, No. 2, 382–401 (1977).

    Google Scholar 

  25. V. M. Zolotarev, “Ideal metrics in the problems of probability theory and mathematical statistics,” Austral. J. Statist.,21, No. 3, 193–208 (1979).

    Google Scholar 

  26. D. Stoyan, “Über einige Eigenschaften monotoner stochastischer Prozesse,” Math. Nachr.,52, 21–34 (1972).

    Google Scholar 

  27. D. Stoyan, “Ein Stetigkeitssatz für einlinige Wartesysteme der Bedienungstheorie,” Math. Operationsforsch. u. Statist.,3, 103–111 (1972).

    Google Scholar 

  28. E. Lukacs, “Stability theorems for characterization by constant regression,” Period. Math. Hung.,2, Nos. 1–4, 111–128 (1972).

    Google Scholar 

  29. S. Beer and E. Lukacs, “Stability theorems for a characterization of the Poisson distribution,” Teor. Veroyatn. Primen.,19, No. 4, 689–699 (1974).

    Google Scholar 

  30. V. M. Zolotarev, “A sharpening of the inequality of Berry-Esseen,” Z. Wahrsch. Verw. Gebiete,8, No. 4, 332–342 (1967).

    Google Scholar 

  31. R. V. Yanushkyavichyus, “Estimates of stability of the characterization of the normal law by identical distribution of a monomial and a linear statistic,” Litov. Mat. Sb.,20, No. 2, 195–206 (1980).

    Google Scholar 

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Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 145–154.

In conclusion, the author uses the opportunity to express his profound gratitude to V. M. Zolotarev for his invariable benevolence and support in the work.

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Shiganov, I.S. A metric approach to investigation of the stability of Pólya theorem on characterization of the normal distribution. J Math Sci 34, 1569–1577 (1986). https://doi.org/10.1007/BF01089799

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