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Stability theorems

  • Eugene Lukacs (a1)

Abstract

A stability theorem determines the extent to which the conclusions of a given theorem are affected if the assumptions of the theorem are not exactly but only approximately satisfied. The meaning of the word ‘approximately’ has to be defined exactly. The stability of decomposition theorems, of characterizations by independence and by regression properties are the primary object of the paper.

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[1] Azlarov, T. A. (1972) Stability of characterizing properties of the exponential distribution (Russian). Litovsk Mat. Sb. 12, 59.
[2] Azlarov, T. A., Dzamirzaev, A. A. and Sultanova, M. M. (1972) Characterizing properties of the exponential distribution and their stability (Russian). Random Processes and Statistical Inference II, Izdat. Fan. Tashkent, 1019.
[3] Bednarek-Kozek, B. and Kozek, A. (1972) On the robustness of properties characterizing the normal distribution. Zast. Mat. (Applicationes Mathematicae) 13, 215230.
[4] Beer, S. and Lukacs, E. (1971) On the robustness of the distribution of a quotient. 38th Session, Internat. Statist. Inst., Washington D.C., Contributed Papers, 3235.
[5] Beer, S. and Lukacs, E. (1972) Remarks concerning a stability theorem for a characterization of the Gamma distribution. Period. Math. Hungar. 2, 129132.
[6] Beer, S. and Lukacs, E. (1974) Stability theorems for a characterization of the Poisson distribution. Teor. Verojatnost. i Primenen 19, 689699; Theor. Prob. Appl. 19 (1974), 656–668.
[7] Bernstein, S. N. (1941) Sur une propriété de la loi de Gauss. Trudy Leningrad. Technolog. Inst. Holod. Promysl. 3, 2122.
[8] čistjakov, G. P. (1969) Stability for the theorem of Yu. V. Linnik. Teor. Funkcii Funkcional. Anal. i Prilozen. 9, 118133.
[9] Cramér, H. (1936) Über eine Eigenschaft der normalen Verteilungsfunktion. Math. Z. 41, 405414.
[10] Cuppens, R. (1963) Sur la stabilité des décompositions en arithmétique des lois de probabilité. C.R. Acad. Sci. Paris 256, 35603561.
[11] Cuppens, R. (1968) Sur la notion de stabilité en arithmétique des lois de probabilité. C.R. Acad. Sci. Paris A 267, 10001002.
[12] Esseen, C. G. (1944) Fourier analysis of distribution functions. Acta Math. 77, 1125.
[13] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.
[14] Gabovic, Yu. R. (1972) The effect of the stability of the characterization of the normal distribution in a theorem of S. N. Bernstein. Dokl. Akad. Nauk SSSR 205, 315. English translation: Soviet Math. Dokl. 13 (1972), 861–863.
[15] Gabovic, Yu. R. (1974) On the stability of certain characteristic properties of the normal distribution. Teor. Verojatnost. i Primenen. 19, 374382. English translation: Theor. Prob. Appl. 19 (1974), 365–371.
[16] Geary, R. C. (1936) Distribution of Student's ratio in nonnormal samples. J. R. Statist. Soc. Suppl. B 3, 178184.
[17] Nhu, Hoang Huu (1966) On the stability of certain theorems characterizing the normal distribution. Internat. Congr. Math. Moscow 1966, Inf. Bull. 6.
[18] Nhu, Hoang Huu (1968) The estimation of the stability of a characterization of the exponential distribution. Litovsk. Mat. Sb. 8, 175177. English translation: Selected Trans. Math. Statist. Prob. 11 (1973), 216-218.
[19] Nhu, Hoang Huu (1968) On the stability of certain characterizations of a normal population. Teor. Verojatnost. i Primenen. 13, 308314. English translation: Theory Prob. Appl. 13 (1968), 299–304.
[20] Kallenberg, O. (1972) Stability in the decomposition of probability measures with finite support. Z. Wahrscheinlichkeitsth. 23, 216223.
[21] Kawata, T. and Sakamoto, H. (1949) On the characterization of the normal population by the independence of the sample mean and the sample variance. J. Math. Soc. Japan 1, 111115.
[22] Laha, R. G. and Lukacs, E. (1960) On a problem connected with quadratic regression. Biometrika 47, 335343.
[23] Laha, R. G. and Lukacs, E. (1960) On a characterization of the Wiener process. Trans. 2nd Prague Conf., Czechoslovak Academy of Sciences, Prague.
[24] Lévy, P. (1937) Théorie de l'Addition des Variables Aléatoires. Gauthier-Villars, Paris.
[25] Linnik, Yu. V. (1957) On the factorization of compositions of Gauss and Poisson laws. Teor. Verojatnost. i Primenen. 2, 3459. English translation: Theory Prob. Appl. 2 (1957), 31–57.
[26] Linnik, Yu. V. (1960) Razlozenie verojatnostih zakonov. Izdat. Leningrad. Univ. English translation: Decomposition of Probability Laws (1964), Oliver and Boyd, Edinburgh and London.
[27] Lukacs, E. (1942) A characterization of the normal distribution. Ann. Math. Statist. 13, 9193.
[28] Lukacs, E. (1961) On the characterization of a family of populations which includes the Poisson population. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 3–4, 159175.
[29] Lukacs, E. and Laha, R. G. (1964) Applications of Characteristic Functions. Griffin, London.
[30] Lukacs, E. (1970) Characteristic Functions, 2nd edn. Griffin, London.
[31] Lukacs, E. (1972) Stability theorems for characterization by constant regression. Period. Math. Hungar. 2, 111128.
[32] Lukacs, E. (1976) Über die Stabilität einer Charakterisierung der Normalverteilung. Sitzungsber. Öst. Akad, Wiss.
[33] Lukacs, E. (1977) A stability theorem for a characterization of the Wiener distribution. Transactions of the 7th Prague Conference. To appear.
[34] Mačis, Ju. Ju. (1967) On the stability of Raikov's theorem (Russian) Litovsk Mat. Sb. 14, 123127.
[35] Mačis, Ju. Ju. (1969) On the stability of the decomposition of the unit distribution law. Teor. Verojatnost. i Primenen. 14, 715718. English translation: Theory Prob. Appl. 14 (1969), 688–690.
[36] Macis, Ju. Ju. (1971) Estimates of the stability theorems for decompositions of the Poisson distribution. Teor. Verojatnost. i Primenen. 16, 218228. English translation: Theory Prob. Appl. 16 (1971), 215–227.
[37] Mačis, Ju. Ju. (1973) On the stability of decompositions of the two point distribution (Russian). Litovsk Mat. Sb. 13, 131138.
[38] Mačis, Ju. Ju. (1974) On the problem of quantitative stability of the decomposition of the binomial law (Russian). Litovsk Mat. Sb. 14, 123127.
[39] Maloshevskii, S. G. (1967) A distribution function whose moments are close to those of a normal law. Mat. Zametki 2, 627633. English translation: Math. Notes 2 (1967), 870–874.
[40] Maloshevskii, S. G. (1968) Sharpness of an estimate of N. A. Sapogov on the stability problem of Cramér's theorem. Teor. Verojatnost. i Primenen. 13, 522525. English translation: Theory Prob. Appl. 13 (1968) 494–496.
[41] Meshalkin, L. D. (1968) On the robustness of some characterizations of the normal distribution. Ann. Math. Statist. 39, 17471750.
[42] Nhu, Hoang Huu, see Nhu, Hoang Huu.
[43] Ostrovskii, I. V. (1965) On the factorization of the convolution of laws of Gauss and Poisson (Russian). Uspehi Mat. Nauk 20, 166171.
[44] Raikov, D. A. (1938) On the decomposition of Gauss and Poisson laws (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 2, 91124.
[45] Ramachandran, B. (1963) A stability theorem for the binomial law. Sankhyā A 25, 8590.
[46] Rao, C. R. (1967) On some characterizations of the normal law. Sankhyā A 29, 114.
[47] Šalaevskii, O. V. (1959) Stability in Raikov's theorem. Vestnik Leningrad Univ. Mat. Meh. Astronom. 14, 4149. English translation: Selected Trans. Math. Statist. Prob. 4 (1963), 233–244.
[48] Sapogov, N. A. (1950) On a property of the Gaussian distribution law. Dokl. Akad. Nauk SSSR NS 73, 461462.
[49] Sapogov, N. A. (1951) The stability problem for a theorem of Cramér. Izv. Akad. Nauk SSSR Ser. Mat. 15, 205218. English translation: Selected Trans. Math. Statist. Prob. 1 (1961), 41–53.
[50] Sapogov, N. A. (1955) The problem of stability for a theorem of Cramér. Vestnik Leningrad Univ. Mat. Meh. Astronom. 10, 6164.
[51] Sapogov, N. A. (1959) On independent terms of a sum of random variables which is distributed almost normally. Vestnik Leningrad Univ. Mat. Meh. Astronom. 14, 78105. English translation: Selected Trans. Math. Statist. Prob. 5 (1965), 1–31.
[52] Zinger, A. A. (1951) On independent samples from a normal population (Russian). Uspehi Mat. Nauk NS 6, 172175.
[53] Zolotarev, V. M. (1968) On the problem of stability of the decomposition of the normal law into components. Teor. Verojatnost. i Primenen. 13, 738742. English translation: Theory Prob. Appl. 13 (1968) 697–700.
[54] Zolotarev, V. M. (1970) Some new inequalities in probability connected with the Lévy metric. Dokl. Akad. Nauk SSSR 190, 10191021. English translation: Soviet Math. Dokl. 11 (1970) 231–234.
[55] Zolotarev, V. M. (1971) Estimates of the difference between distributions in the Lévy metric. Trudy Steklov 112, 224231. English translation: Proc. Steklov Inst. 112 (1971), 232–240.
* No English translation of these Russian papers was available at the time this paper was written.

Keywords

Stability theorems

  • Eugene Lukacs (a1)

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