Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T03:03:31.942Z Has data issue: false hasContentIssue false
  • English
  • Français

A survey of the theory of characteristic functions

Published online by Cambridge University Press:  01 July 2016

Eugene Lukacs
Eugene Lukacs*
Affiliation:
The Catholic University of America, Washington, D. C.
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper gives a survey of the theory of univariate characteristic functions. These functions were originally introduced as tools in the study of limit theorems but it was later realized that they had an independent mathematical interest. Those parts of the theory which can be found in textbooks are treated only briefly; the main emphasis is placed on more recent developments and areas where active research is still in progress.

Keywords

UNIQUENESS, INVERSION, CONVOLUTION AND CONTINUITY THEOREMSINFINITE CONVOLUTIONANALYTIC AND BOUNDARY C. F.'SDECOMPOSMON OF C. F.'SINDECOMPOSABLE C.F.'SINFINITELY DIVISIBLE C.F.'SSTABLE LAWSTHE CLASS 10ALPHA DECOMPOSITIONSCHARACTERIZATION PROBLEMSSTABILITY THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 1 , April 1972 , pp. 1 - 37
Copyright
Copyright © Applied Probability Trust 1972 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References

A. Text books on probability theory and analysis

[A1] Chung, K. L. (1968) A Course in Probability Theory. Harcourt, Brace and World, New York.Google Scholar
[A2] Cramér, H. (1946) Mathematical Methods of Statistics. Princeton University Press, Princeton, New Jersey.Google Scholar
[A3] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II. J. Wiley, New York.Google Scholar
[A4] Gnedenko, B. V. (1967) The Theory of Probability. Chelsea, New York.Google Scholar
[A5] Gnedenko, B. V. and Kolmogrov, A. N. (1954) Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, Mass. Google Scholar
[A6] Hille, E. (1962) Analytic Function Theory, Vol. II. Ginn and Co., Boston, Mass. Google Scholar
[A7] Loève, M. (1963) Probability Theory. 3rd Ed. Van Nostrand, New York.Google Scholar
[A8] Markushevich, A. I. (1965) Theory of Functions of a Complex Variable, Vol. II. Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
[B1] Dugué, D. (1957) Arithmétique des lois de probabilités. Mém. Sci. Math. 137. Gauthier-Villars, Paris.Google Scholar
[B2] Linnik, Yu. V. (1964) Decomposition of Probability Distributions. Oliver and Boyd, Edinburgh and London.Google Scholar
[B3] Lukacs, E. (1970) Characteristic Functions. 2nd rev. Ed. Griffin, London.Google Scholar
[B4] Lukacs, E. and Laha, R. G. (1964) Applications of Characteristic Functions. Griffin, London.Google Scholar
[B5] Ramachandran, B. (1967) Advanced Theory of Characteristic Functions. Statistical Publishing Society, Calcutta.Google Scholar
Akutowicz, E. J. (1959) On extrapolating a positive definite function from a finite interval. Math. Scand. 7, 157–169.Google Scholar
Akutowicz, E. J. (1960) Sur l'approximation par certaines fonctions entières. Ann. Eccle Norm. Sup. 77, 281–301.Google Scholar
Bergström, H. (1952) On some expansions of stable distributions. Ark. Mat. 2, 375–378.Google Scholar
Bernstein, S. N. (1941) On a property characterizing Gauss' law. Trudi Leningrad. Politehniceskogo Instituta 3, 21–22.Google Scholar
Blum, J. R. and Rosenblatt, M. (1959) On the structure of infinitely divisible distributions. Pacific J. Math. 9, 1–7.Google Scholar
Boas, R. P. (1967) Lipschitz behaviour and integrability of characteristic functions. Ann. Math. Statist. 38, 32–36.Google Scholar
Cuppens, R. (1963) Sur la stabilité des décompositions en arithmétique des lois de probabilité. C. R. Acad Sci. Paris, 265, 3560–3561.Google Scholar
Cuppens, R. (1968) Sur la notion de stabilité en arithmétique des lois de probabilité. C. R. Acad. Sci. Paris, 267, 1000–1002.Google Scholar
Cuppens, R. and Lukacs, E. (1970) On the domain of definition of analytic characteristic functions. Ann. Math. Statist. 41, 1096–1101; correction 42, 415.Google Scholar
Dugué, D. (1951) Analycité et convexité des fonctions caractéristiques. Ann. Inst. Poincaré 12, 45–56.Google Scholar
Dugué, D. and Girault, M. (1955) Fonctions convexes de Pólya. Publ. Inst. Statist. Univ. Paris 4, 3–10.Google Scholar
Esseen, C. G. (1945) Fourier analysis of distribution functions. Acta Math. 77, 1–125.Google Scholar
Esseen, C. G. (1965) On infinitely divisible one-sided distributions. Math. Scand. 17, 65–76.Google Scholar
Feller, W. (1952) On a generalization of Marcel Riesz' potentials and the semi-groups generated by them. Medd. Lunds. Univ. Mat. Sem. Tome Supplémental Marcel Riesz, 73–81.Google Scholar
Gil-Pelaez, J. (1951) Note on the inversion theorem. Biometrika 38, 481–482.Google Scholar
Goldberg, A. A. and Ostrovskii, I.V. (1967) An application of a theorem of U.K. Hayman to a problem in the theory of the decomposition of probability laws, Ukrain. Mat. Z. 19, 104–106.Google Scholar
Hartmann, P. and Wintner, A. (1942) On the infinitesimal generators of integral convolutions. Amer. J. Math. 64, 273–298.Google Scholar
Jessen, B. and Wintner, A. (1935) Distribution functions and the Riemann zeta function. Trans. Amer. Math. Soc. 38, 48–88.Google Scholar
Johansen, S. (1966) An application of extreme point methods to the representation of infinitely divisible distributions. Z. Wahrscheinlichkeitsth. 5, 304–316.Google Scholar
Kawata, T. (1940) On the division of a probability law. Proc. Imp. Acad. Tokyo 16, 249–254.Google Scholar
Kershner, R. and Wintner, A. (1935) On symmetric Bernoulli convolutions. Amer. J. Math. 57, 541–545.Google Scholar
Kershner, R. (1936) On singular Fourier transforms. Amer. J. Math. 58, 450–453.Google Scholar
Khinchine, A. Ya. (1937) Contribution à l'arithmétique des lois de distributions. Bull. Univ. Etat. Moscou Ser. Internationale, A. 1, 6–17.Google Scholar
Krein, M. G. (1940) Sur le problème du prolongement des fonctions hermitiennes, positives et continues. C. R. (Dokl.) Acad. Sci. USSR. 26, 17–20.Google Scholar
Krein, M. G. (1943) On the representation of functions by Fourier-Stieltjes integrals. (In Russian) Učen. Zap. Kubisevs. Gos. Ped. Ucitels. Inst. im V. V. Kuibiseva 7, 123–148.Google Scholar
Lévy, P. (1937) Sur les exponentielles de polynômes. Ann. Ecole Norm. Sup. 73, 231–292.Google Scholar
Lévy, P. (1952) Sur une classe de lois de probabilité indécomposables. C. R. Acad. Sci. Paris 235, 489–491.Google Scholar
Lévy, P. (1961) Quelques problèmes non résolus de la théorie des fonctions caractéristiques. Ann. Mat. Pura Appl. 53, 315–332.Google Scholar
Lewis, T. (1967) The factorization of the rectangular distribution. J. Appl. Prob. 4, 529–542.Google Scholar
Linnik, Yu. V. (1957) On factorizing the composition of a Gaussian and a Poissonian law. Theor. Probability Appl. 2, 31–57.Google Scholar
Lukacs, E. (1956) Characterization of populations by properties of suitable statistics. Proc. Third Berkeley Symp. Math. Statist. Prob. II, 195–214.Google Scholar
Lukacs, E. (1961) Recent developments in the theory of characteristic functions. Proc. Fourth Berkeley Symp. Math. Statist. Prob. II, 307–335.Google Scholar
Lukacs, E. (1963) Applications of characteristic functions in statistics. Sankya A 25, 175–188.Google Scholar
Lukacs, E. (1964) Inversion formulae for characteristic functions of absolutely continuous distributions. Amer. Math. Monthly 71, 44–47; correction ibid 781.Google Scholar
Lukacs, E. (1969a) Some recent development in the theory of multivariate characteristic functions. Second Symposium on Multivariate Analysis. Academic Press, New York.Google Scholar
Lukacs, E. (1969b) Stable distributions and their characteristic functions. Jber. Deutsch. Mathver. 71, 84–114.Google Scholar
Macis, Yu. Yu. (1968) The stability of a theorem of O. A. Shalayevsky. Litovsk. Mat. Sb. 8.Google Scholar
Malosevskh, S. C. (1968) Unimprovability of the result due to N. A. Sapogov in the stability problem of Cramér's theorem. Teor. Veroyat. Primen. 13, 522–525.Google Scholar
Marcinkiewcz, J. (1938) Sur les fonctions indépendentes III. Fund. Math. 31, 86–102.Google Scholar
Meshalkin, L. D. (1968) On the robustness of some characterizations of the normal distribution. Ann. Math. Statist. 39, 1747–1750.Google Scholar
Hoang Huu Nhu (Hiu Nye) (1966) On the stability of some characterizations for the normal population. Abstract of a paper presented at the International Congress of Mathematicians, Moscow 1966, Information Bulletin No. 6.Google Scholar
Hoang Huu Nhu (Hiu Nye) (1968) On the stability of some characterizations for the normal population. Teor. Veroyat. Primen. 13, 308–314.Google Scholar
Ostrovskii, I. V. (1963) Entire functions satisfying some special inequalities connected with the theory of characteristic functions of probability laws. (In Russian) Har'kov. Univ. Mat. Obshestva 29, 145–168. Translated in AMS-IMS. Selected Transl. Math. Statist. Prob. 7, 203–234.Google Scholar
Ostrovskii, I. V. (1964) On the decomposition of infinitely divisible lattice laws. (In Russian) Vestnik Leningrad. Univ. 19, 51–60.Google Scholar
Ostrovskii, I. V. (1965) Some theorems on decompositions of probability laws. (In Russian) Proc. Steklov Institute Math. 19, 198–235. Translated in AMS (1966) 221–259.Google Scholar
Ostrovskii, I. V. (1966) On the factorization of multidimensional infinitely divisible probability laws without Gaussian component. Vestnik Har'kov. Gos. Univ. (Ser. Mat. Meh.) 32, 51–72.Google Scholar
Pitman, E. J. G. (1956) On the derivation of a characteristic function at the origin. Ann. Math. Statist. 27, 1156–1160.Google Scholar
Raikov, D. A. (1938) On the decomposition of Gauss and Poisson laws. Izv. Akad. Nauk SSSR (Ser. Mat.) 2, 91–124.Google Scholar
Raikov, D. A. (1940) On positive definite functions. (In Russian) Dokl. Akad. Nauk SSSR 26, 857–862.Google Scholar
Ramachandran, B. (1966) On one-sided distribution functions. Sankya A 28, 315–318.Google Scholar
Rossberg, H. J. (1966) Wachstum and Nullstellenverteilung ganzer charakteristischer Funktionen. Monatsb. Deutsch. Akad. Wiss. Berlin 8, 275–286.Google Scholar
Sapogov, N. A. (1951) The stability problems for a theorem of Cramér. (In Russian) Izv. Akad. Nauk SSSR. (Ser. Mat.) 15, 205–218. Translated in AMS-IMS Selected Transl. Math. Statist. Prob. 1, 41–53.Google Scholar
Schwartz, L. (1941) Sur le module de la fonction caractéristique du calcul des probabilités. C. R. Acad. Sci. Paris 212, 418–421.Google Scholar
Shalayevsky, O. V. (1959) On the stability of a theorem of D. A. Raikov. (In Russian) Vestnik Leningrad. Univ. 14, 41–49. Translated in AMS-IMS Selected Transl. Math. Statist. Prob. 4, 233–244.Google Scholar
Shimizu, R. (1964) On the decomposition of infinitely divisible characteristic functions with a continous Poisson spectrum. Ann. Inst. Statist. Math. 16, 387–407.CrossRefGoogle Scholar
Tucker, H. (1965) On a necessary and sufficient condition that an infinitely divisible distribution be absolutely continuous. Trans. Amer. Math. Soc. 118, 316–330.Google Scholar
Wintner, A. (1938) Asymptotic distributions and infinite convolutions. (Planographed lecture notes) Edward Brothers, Ann Arbor, Michigan.Google Scholar
Wintner, A. (1947) The Fourier transforms of probability distributions. (Mimeographed lecture notes, published by the author.) Google Scholar
Wintner, A. (1956) Cauchy's stable distributions and an “explicit formula” of Mellin. Amer. J. Math. 78, 819–871.Google Scholar
Zolotarev, V. M. (1963) Analytical structure of the infinitely divisible laws of the L-class. Litovsk. Mat. Sb. 3, 123–140.Google Scholar
Zolotarev, V. M. (1968) On the stability of a normal law decomposition in components. Teor. Veroyat. Primen. 13, 738–742.Google Scholar
Zygmund, A. (1947) A remark on characteristic functions. Ann. Math. Statist. 18, 272–276.Google Scholar