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Analytical methods in probability theory

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Symposium on Probability Methods in Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 31))

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References

  1. S. Bochner, Vorlesungen über Fouriersche Integrale. Akad. Verlagsgesellschaft, Leipzig (1932) reprinted by Chelsea Publ.Co.New York (1948).

    Google Scholar 

  2. R. Cairoli, Sur les fonctions caractéristiques des lois de probabilité. Publ.Inst.Stat.Univ. Paris 13, 45–53 (1964).

    MathSciNet  MATH  Google Scholar 

  3. I.F. Christensen, Some further extensions of a theorem of Marcinkiewicz. Pacific J. Math. 12, 59–67 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Cramér, On the representation of a function by certain Fourier integrals. Trans. Amer. Math.Soc. 46, 191–201 (1939).

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Cuppens, Sur la décomposition d'une fonction 2q fois dérivable à l'origine en produit infini des fonctions caractéristiques. C.R.Acad.Sci., Paris 256, 3806–3808 (1963)

    MathSciNet  MATH  Google Scholar 

  6. G. Darmois, Sur une propriété caractéristique de la loi de Gauss. C.R. Acad. Sci. Paris, 232, 1999–2000 (1951).

    MathSciNet  MATH  Google Scholar 

  7. G. Darmois, Analyse générale des liaisons stochastiques. Revue Intern. Stat.Inst. 21, 2–8, (1953).

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Dugué, Arithmétique des lois de probabilité. Mémorial des Sciences Mathématiques 137, Gauthier-Villars, Paris (1957).

    MATH  Google Scholar 

  9. D. Dugué, Résultats sur les fonctions absolument monotones et applications à l'arithmétique des fonctions de type positif. C.R. Acad. Sci. Paris 244, 715–717 (1957).

    MathSciNet  MATH  Google Scholar 

  10. D. Dugué, Sur le Théorème de Lévy-Cramér. Publ.Inst.Stat.Univ. Paris 6, 213–225 (1957).

    MATH  Google Scholar 

  11. M. Girault, Analycité et periodicité des fonctions caractéristiques. Publ.Inst. Stat. Univ. Paris 5, 91–94, (1956).

    MathSciNet  MATH  Google Scholar 

  12. B.V. Gnedenko, Sur les fonctions caractéristiques. Bull.Univ. d'Etat à Moscou (Serie internationale) 1, 1–3 (1937).

    MathSciNet  MATH  Google Scholar 

  13. B.V. Gnedenko, The theory of probability. Chelsea Publ. Co., New York (1962).

    MATH  Google Scholar 

  14. U. Grenander, Probabilities on algebraic structures. J.Wiley and Sons,New York (1963)

    MATH  Google Scholar 

  15. C.S. Herz, Fonctions opérant sur les fonctions définies positives. Annales Inst. Fourier, Grenoble 13, 161–180 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Hille, Analytic function theory, vol. II. Ginn and Co., Boston, Mass. (1962).

    MATH  Google Scholar 

  17. A.I. Ingham, A note on Fourier transforms. J.London Math. Soc. 9, 27–32 (1936)

    MathSciNet  Google Scholar 

  18. T. Kawata, On the division of a probability law. Proc.Imp.Acad. Tokyo, 16, 249–254 (1940).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Khinchine, Zur Kennzeichnung der charakteristischen Funktionen. Bull. Univ. d'Etat à Moscou (Serie internationale), 1, 1–3 (1937).

    Google Scholar 

  20. M.G. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues. C.R. (Doklady) Ac. Sci. USSR 26, 17–20 (1940).

    MathSciNet  MATH  Google Scholar 

  21. R.G. Laha, On a factorization theorem in the theory of analytic characteristic functions. Ann. Math. Stat. 29, 922–926 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  22. R.G. Laha, On functions having non-negative derivatives at the origin. Proc.Amer. Math. Soc. 12, 175–180 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  23. R.G. Laha-E. Lukacs, On a factorization of characteristic functions which have a finite number of derivatives at the origin. Publ.Inst. Stat. Univ. Paris 11, 221–224 (1962).

    MathSciNet  MATH  Google Scholar 

  24. R.G. Laha-E. Lukacs-A. Rényi, A generalization of a theorem of E. Vincze. Vincze. Publ.Math. Inst. Hungarian Acad. Sci. IX, section A, 237–239 (1964).

    MathSciNet  MATH  Google Scholar 

  25. R.G. Laha-E. Lukacs, On linear forms whose distribution is identical with that of a monomial. Pacific J.Math, 15, 207–214 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  26. N. Levinson, On a class of non-vanishing functions. Proc. London Math. Soc. 41, 393–407 (1936).

    Article  MathSciNet  MATH  Google Scholar 

  27. N. Levinson, A theorem relating non-vanishing and analytic functions. J.Math.Phys. 16, 185–190 (1938).

    Article  MATH  Google Scholar 

  28. P. Lévy, Sur les exponentielles de polynômes. Annales Ecole Normale Supérieure 73, 231–292 (1937).

    MATH  Google Scholar 

  29. Yu.V. Linnik, Linear forms and statistical criteria I, II, Ukrainski mat.zurnal 5, 207–243 and 247–290 (1953). (English translation: Selected translations in Math. Stat. and Probability 3, 1–90, Amer. Math.Soc., Providence, R.I.).

    MathSciNet  MATH  Google Scholar 

  30. Yu.V. Linnik, On polynomial statistics in connection with the analytical theory of differential equations. Vestnik Leningrad Univ. 11, 35–48 (1956). (Engl. translations: Selected in Math. Statistics and Probability 1, 191–206, Amer. Math. Soc., Providence, R.I.).

    MathSciNet  MATH  Google Scholar 

  31. Yu.V. Linnik, A remark on the decomposition of the Normal law. Teoriya Veroyatn. i primen. 1, 435–436 (1956).

    MathSciNet  Google Scholar 

  32. Yu.V. Linnik, On factorizing the composition of a Gaussian and a Poissonian law. Teoriya Veroyatn. i Primen. 2, 34–59 (1957).

    MathSciNet  MATH  Google Scholar 

  33. Yu.V. Linnik, General theorems in the factorization of infinitely divisible laws. I, Three basic lemmas, necessary conditions. Teoriya Veroyatn i Primen. 3, 3–40 (1958).

    MathSciNet  Google Scholar 

  34. Yu.V. Linnik, General theorems on the factorization of infinitely divisible laws II. Sufficient conditions. Teoriya Veroyatn. i.Primen. 4, 55–85 (1959).

    MathSciNet  Google Scholar 

  35. Yu.V. Linnik, General theorems on the factorization of infinitely divisible laws III. Teor.Veroyatn. i Primen. 4, 150–171, (1959).

    Google Scholar 

  36. Yu.V. Linnik, On α-decompositions of infinitely divisible probability laws. Vestnik Leningrad Univ. 1, 14–23 (1959).

    MathSciNet  Google Scholar 

  37. Yu.V. Linnik, Decomposition of probability distributions. Oliver & Boyd, London (1964). (Russian original published in 1960).

    MATH  Google Scholar 

  38. Yu.V. Linnik-V.P. Skitovich, Again on the generalization of Cramér's theorem. Vestnik Leningrad Univ. 13, 39–44 (1958).

    MathSciNet  MATH  Google Scholar 

  39. E. Lukacs, Characterization of populations by properties of suitable statistics. Proc. Third Berkeley symposium 2, 195–214 (1956). Univ. of Calif. Press, Berkeley, California.

    MathSciNet  MATH  Google Scholar 

  40. E. Lukacs, Characteristic functions. Ch. Griffin & Co., London (1960).

    MATH  Google Scholar 

  41. E. Lukacs, Recent developments in the theory of characteristic functions. Proc. Fourth Berkeley Symposium 2, 307–335, (1961) Univ. of Calif. Press, Berkeley, California.

    MathSciNet  MATH  Google Scholar 

  42. L.V. Mamay, On the theory of characteristic functions. Vestnik Leningrad Univ. 15 (1960) 85–99. (English translation: Selected Translations Math. Stat. and Probability 4, 153–170, Amer. Math. Soc., Providence, R.I.).

    MathSciNet  Google Scholar 

  43. J. Marcinkiewicz, Sur une propriété de la loi de Gauss. Math. Zeitschr. 44, 622–638 (1938).

    MathSciNet  MATH  Google Scholar 

  44. M. Mathias, Über positive Fourier Integrale. Math. Zeitschr. 16, 103–125 (1923).

    Article  MathSciNet  MATH  Google Scholar 

  45. H.D. Miller, Generalization of a theorem of Marcinkiewicz. To be published in Pacific J. of Math.

    Google Scholar 

  46. I.V. Ostrovskii, On the application of a law established by Wiman and Valiron to the investigation of characteristic functions of probability laws. Doklady Akad. Nauk SSSR, 143, 532–535 (1962) (English Translation: Soviet Mathematics, Doklady 3, 532–535).

    MathSciNet  Google Scholar 

  47. I.V. Ostrovskii, On the decomposition of the convolution of Gauss and Poisson laws. Uspeki Mat. Nauk 20 (124) 166–171 (1965).

    MathSciNet  Google Scholar 

  48. I.V. Ostrovskii, Some theorems on the decomposition of probability laws. Trudi Mat. Inst. im. V.A. Steklov 79, 198–235 (1965).

    MathSciNet  Google Scholar 

  49. G. Pólya, Remarks on characteristic functions. Proceedings First Berkeley Symposium 115–123, Univ. of California Press, Berkeley-Los Angeles 1949.

    Google Scholar 

  50. D.A. Raikov, On the decomposition of Gauss and Poisson laws. Izv. Akad. Nauk SSSR (Ser. Mat.) 2, 91–124 (1938).

    MATH  Google Scholar 

  51. B. Ramachandran, Application of a theorem of Mamay's to a “denumerable α-decomposition” of the Poisson law.Publ. Inst. Stat. Univ. Paris 13, 13–19 (1964).

    MathSciNet  MATH  Google Scholar 

  52. W. Rudin, Fourier Analysis on Groups, Wiley and Sons, New York (1962).

    MATH  Google Scholar 

  53. V.P. Skitovich, On a property of the normal distribution. Dokl. Akad. Nauk SSSR 89, 217–219 (1953).

    MathSciNet  Google Scholar 

  54. V.P. Skitovich, Linear forms in independent random variables and the normal distribution. Izv. Akad. Nauk SSSR (Ser. Mat.) 18, 185–200 (1954). (English translation: Selected Translations Math. Stat. and Probability 2, 211–228, Amer. Math. Soc., Providence, R.I.).

    Google Scholar 

  55. A.V. Skorohod, On a theorem relative to stable distributions. Uspeki Mat. Nauk (NS) 9 (60) 189–190 (1954).

    MathSciNet  Google Scholar 

  56. A.V. Skorohod, Asymtotic formulas for stable distribution laws. Dokl.Akad. Nauk SSSR 98, 731–734 (1954).

    MathSciNet  Google Scholar 

  57. H. Teicher, Sur les puissances des fonctions caractéristiques. C.R. Acad.Sci. Paris 246, 694–696 (1958).

    MathSciNet  MATH  Google Scholar 

  58. E. Vincze, Bemerkung zur Charakterisierung des Gausschen Fehlergesetzes. Publ. Math. Inst. Hungarian Acad. Sci. 7, 357–361 (1962).

    MathSciNet  MATH  Google Scholar 

  59. A.A. Zinger-Yu.V. Linnik, On a class of differential equations with an application to certain questions of regression theory. Vestnik Leningrad Univ. 12, 121–130 (1955). (English Translation: Selected Translations Mathematical Statistics and Probability 3, 181–190. Amer. Math.Soc., Providence, R.I.).

    MathSciNet  MATH  Google Scholar 

  60. A.A. Zinger-Yu.V. Linnik, On an analytic extension of a theorem of Cramér and its application. Vestnik Leningrad Univ. 10, 51–56 (1955).

    MathSciNet  Google Scholar 

  61. V.M. Zolotarev, Expression of the density of a stable distribution with exponent α > 1 by means of a density with exponent 1/α. Dokl.Akad. Nauk SSSR, (NS) 98, 735–738 (1954). (English Translation: Selected Translations Mathematical Statistics and Probability 1, 163–167, Amer. Math. Soc., Providence, R.I., (1961)).

    MathSciNet  Google Scholar 

  62. V.M. Zolotarev, Mellin-Stieltjes transforms in probability theory. Teoriya Veroyatn.i Primen. 2, 444–469 (1957).

    MathSciNet  MATH  Google Scholar 

  63. V.M. Zolotarev, On the representation of stable laws by integrals. Trudi mat. inst. im. Steklova 71, 46–50 (1964).

    MathSciNet  Google Scholar 

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Authors
  1. Eugene Lukacs
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© 1967 Springer-Verlag

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Lukacs, E. (1967). Analytical methods in probability theory. In: Symposium on Probability Methods in Analysis. Lecture Notes in Mathematics, vol 31. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0061120

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  • DOI: https://doi.org/10.1007/BFb0061120

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  • Print ISBN: 978-3-540-03902-0

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