Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Bochner, Vorlesungen über Fouriersche Integrale. Akad. Verlagsgesellschaft, Leipzig (1932) reprinted by Chelsea Publ.Co.New York (1948).
R. Cairoli, Sur les fonctions caractéristiques des lois de probabilité. Publ.Inst.Stat.Univ. Paris 13, 45–53 (1964).
I.F. Christensen, Some further extensions of a theorem of Marcinkiewicz. Pacific J. Math. 12, 59–67 (1962).
H. Cramér, On the representation of a function by certain Fourier integrals. Trans. Amer. Math.Soc. 46, 191–201 (1939).
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)
G. Darmois, Sur une propriété caractéristique de la loi de Gauss. C.R. Acad. Sci. Paris, 232, 1999–2000 (1951).
G. Darmois, Analyse générale des liaisons stochastiques. Revue Intern. Stat.Inst. 21, 2–8, (1953).
D. Dugué, Arithmétique des lois de probabilité. Mémorial des Sciences Mathématiques 137, Gauthier-Villars, Paris (1957).
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).
D. Dugué, Sur le Théorème de Lévy-Cramér. Publ.Inst.Stat.Univ. Paris 6, 213–225 (1957).
M. Girault, Analycité et periodicité des fonctions caractéristiques. Publ.Inst. Stat. Univ. Paris 5, 91–94, (1956).
B.V. Gnedenko, Sur les fonctions caractéristiques. Bull.Univ. d'Etat à Moscou (Serie internationale) 1, 1–3 (1937).
B.V. Gnedenko, The theory of probability. Chelsea Publ. Co., New York (1962).
U. Grenander, Probabilities on algebraic structures. J.Wiley and Sons,New York (1963)
C.S. Herz, Fonctions opérant sur les fonctions définies positives. Annales Inst. Fourier, Grenoble 13, 161–180 (1963).
E. Hille, Analytic function theory, vol. II. Ginn and Co., Boston, Mass. (1962).
A.I. Ingham, A note on Fourier transforms. J.London Math. Soc. 9, 27–32 (1936)
T. Kawata, On the division of a probability law. Proc.Imp.Acad. Tokyo, 16, 249–254 (1940).
A. Khinchine, Zur Kennzeichnung der charakteristischen Funktionen. Bull. Univ. d'Etat à Moscou (Serie internationale), 1, 1–3 (1937).
M.G. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues. C.R. (Doklady) Ac. Sci. USSR 26, 17–20 (1940).
R.G. Laha, On a factorization theorem in the theory of analytic characteristic functions. Ann. Math. Stat. 29, 922–926 (1958).
R.G. Laha, On functions having non-negative derivatives at the origin. Proc.Amer. Math. Soc. 12, 175–180 (1961).
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).
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).
R.G. Laha-E. Lukacs, On linear forms whose distribution is identical with that of a monomial. Pacific J.Math, 15, 207–214 (1965).
N. Levinson, On a class of non-vanishing functions. Proc. London Math. Soc. 41, 393–407 (1936).
N. Levinson, A theorem relating non-vanishing and analytic functions. J.Math.Phys. 16, 185–190 (1938).
P. Lévy, Sur les exponentielles de polynômes. Annales Ecole Normale Supérieure 73, 231–292 (1937).
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.).
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.).
Yu.V. Linnik, A remark on the decomposition of the Normal law. Teoriya Veroyatn. i primen. 1, 435–436 (1956).
Yu.V. Linnik, On factorizing the composition of a Gaussian and a Poissonian law. Teoriya Veroyatn. i Primen. 2, 34–59 (1957).
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).
Yu.V. Linnik, General theorems on the factorization of infinitely divisible laws II. Sufficient conditions. Teoriya Veroyatn. i.Primen. 4, 55–85 (1959).
Yu.V. Linnik, General theorems on the factorization of infinitely divisible laws III. Teor.Veroyatn. i Primen. 4, 150–171, (1959).
Yu.V. Linnik, On α-decompositions of infinitely divisible probability laws. Vestnik Leningrad Univ. 1, 14–23 (1959).
Yu.V. Linnik, Decomposition of probability distributions. Oliver & Boyd, London (1964). (Russian original published in 1960).
Yu.V. Linnik-V.P. Skitovich, Again on the generalization of Cramér's theorem. Vestnik Leningrad Univ. 13, 39–44 (1958).
E. Lukacs, Characterization of populations by properties of suitable statistics. Proc. Third Berkeley symposium 2, 195–214 (1956). Univ. of Calif. Press, Berkeley, California.
E. Lukacs, Characteristic functions. Ch. Griffin & Co., London (1960).
E. Lukacs, Recent developments in the theory of characteristic functions. Proc. Fourth Berkeley Symposium 2, 307–335, (1961) Univ. of Calif. Press, Berkeley, California.
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.).
J. Marcinkiewicz, Sur une propriété de la loi de Gauss. Math. Zeitschr. 44, 622–638 (1938).
M. Mathias, Über positive Fourier Integrale. Math. Zeitschr. 16, 103–125 (1923).
H.D. Miller, Generalization of a theorem of Marcinkiewicz. To be published in Pacific J. of Math.
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).
I.V. Ostrovskii, On the decomposition of the convolution of Gauss and Poisson laws. Uspeki Mat. Nauk 20 (124) 166–171 (1965).
I.V. Ostrovskii, Some theorems on the decomposition of probability laws. Trudi Mat. Inst. im. V.A. Steklov 79, 198–235 (1965).
G. Pólya, Remarks on characteristic functions. Proceedings First Berkeley Symposium 115–123, Univ. of California Press, Berkeley-Los Angeles 1949.
D.A. Raikov, On the decomposition of Gauss and Poisson laws. Izv. Akad. Nauk SSSR (Ser. Mat.) 2, 91–124 (1938).
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).
W. Rudin, Fourier Analysis on Groups, Wiley and Sons, New York (1962).
V.P. Skitovich, On a property of the normal distribution. Dokl. Akad. Nauk SSSR 89, 217–219 (1953).
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.).
A.V. Skorohod, On a theorem relative to stable distributions. Uspeki Mat. Nauk (NS) 9 (60) 189–190 (1954).
A.V. Skorohod, Asymtotic formulas for stable distribution laws. Dokl.Akad. Nauk SSSR 98, 731–734 (1954).
H. Teicher, Sur les puissances des fonctions caractéristiques. C.R. Acad.Sci. Paris 246, 694–696 (1958).
E. Vincze, Bemerkung zur Charakterisierung des Gausschen Fehlergesetzes. Publ. Math. Inst. Hungarian Acad. Sci. 7, 357–361 (1962).
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.).
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).
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)).
V.M. Zolotarev, Mellin-Stieltjes transforms in probability theory. Teoriya Veroyatn.i Primen. 2, 444–469 (1957).
V.M. Zolotarev, On the representation of stable laws by integrals. Trudi mat. inst. im. Steklova 71, 46–50 (1964).
Rights and permissions
Copyright information
© 1967 Springer-Verlag
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/BFb0061120
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-03902-0
Online ISBN: 978-3-540-34970-9
eBook Packages: Springer Book Archive