# Distribution Functions and Characteristic Functions

• Yuan Shih Chow
• Henry Teicher
Part of the Springer Texts in Statistics book series (STS)

## Abstract

Distribution functions are mathematical artifacts with properties that are independent of any probabilistic setting. Notwithstanding, most of the theorems of interest are geared to d.f.s of r.v.s and the majority of proofs are simpler and more intuitive when couched in terms of r.v.s having, or probability measures determined by, the given d.f.s. Since r.v.s possessing preassigned d.f.s can always be defined on some probability space, the language of r.v.s and probability will be utilized in many of the proofs without further ado.

### Keywords

Covariance Convolution Metron

## Preview

### References

1. H. E. Bray, “Elementary properties of the Stieltjes integral,”Ann. Math.20 (1919), 177–186.
2. F. P. Cantelli, “Una teoria astratta del calcola delle probabilitá,”Ist. Ital.Attuari3 (1932).Google Scholar
3. H. Chernoff. “Large sample theory: Parametric caseAnn. Math. Stat.27 (1956), 1–22.
4. Y. S. Chow and H. Robbins, “On optimal stopping rules,”Z. Wahr.2 (1963), 33–49.
5. K. L. ChungA Course in Probability TheoryHarcourt Brace, New York, 1968; 2nd ed., Academic Press, New York, 1974.Google Scholar
6. H. Cramér, ” Über eine Eigenschaft der normalen Verteilungsfunktion,”Math. Z. 41(1936), 405–414.
7. H. CramérMathematical Methods of StatisticsPrinceton Univ. Press, Princeton, 1946.
8. H. CramérRandom Variables and Probability DistributionsCambridge Tracts Math. No. 36, Cambridge Univ. Press, London, 1937; 3rd ed., 1970.
9. J. L. DoobStochastic ProcessesWiley, New York, 1953.
10. W. FellerAn Introduction to Probability Theory and Its ApplicationsVol. 2, Wiley, New York, 1966.
11. M. Fréchet and J. Shohat, “A proof of the generalized second limit theorem in the theory of probability,”Trans. Amer. Math. Soc.33 (1931).Google Scholar
12. J. GlivenkoStieltjes Integral1936 [in Russian].
13. B. V. Gnedenko and A. N. KolmogorovLimit Distributions for Sums of Independent Random Variables(K. L. Chung, translator), Addison-Wesley, Reading, Mass., 1954.Google Scholar
14. G. H. HardyA course of Pure Mathematics10th ed., Cambridge Univ. Press, New York, 1952.
15. E. Helly, “Uber lineare Funktionaloperationen,”Sitz. Nat. Kais. Akad. Wiss. 121IIa (1921), 265–277.Google Scholar
16. P. LévyCalcul des probabilitésGauthier-Villars, Paris, 1925.
17. P. LévyThéorie de l’addition des variables aléatoiresGauthier-Villars, Paris, 1937; 2nd ed., 1954.
18. M. LoèveProbability Theory3rd ed., Van Nostrand, Princeton, 1963; 4th ed., Springer-Verlag, Berlin and New York, 1977–1978.Google Scholar
19. E. LukacsCharacteristic Functions2nd ed., Hoffner, New York, 1970.
20. G. Polya, “ Remarks on characteristic functions,”Proc. 1st Berkeley Symp. Stat. and Prob. 1949 115–123. Google Scholar
21. D. A. Raikov, “On the decomposition of Gauss and Poisson laws,”Izv. Akad. Nauk USSR (Ser. Mat.) 2(1938a), 91–124 [in Russian].Google Scholar
22. D. A. Raikov, “ Un théorème de la théorie des fonctions caracteristiques analytiques,”Izvest. Fak. Mat. Mek. Univ. Tomsk NII2 (1938b), 8–11.Google Scholar
23. H. Robbins, “Mixture of distributions,”Ann. Math. Stat. 19(1948), 360–369.
24. S. SaksTheory of the Integral(L. C. Young, translator), Stechert-Hafner, New York, 1937.Google Scholar
25. N. A. Sapogov, “The stability problem for a theorem of Cramer,”Izv. Akad. Nauk USSR (ser. Mat.) 15(1951), 205–218. [See also selected translationsMath. Stat. and Prob. 141–53, Amer. Math. Soc.]
26. H. Scheffé, “A useful convergence theorem for probability distributions,”Ann. Math. Stat. 18(1947), 434–438.
27. J. A. Shohat and J. D. Tamarkin, “The problem of moments,” Math. Survey No. 1, Amer. Math. Soc., New York, 1943.Google Scholar
28. E. Slutsky, “Uber stochastiche Asymptoten und Grenzwerte,” Metron5(1925), 1–90.Google Scholar
29. H. Teicher, “On the factorization of distributions,” Ph.D. Thesis, 1950. [See alsoAnn. Math. Stat. 25(1954), 769–774.]Google Scholar
30. H. Teicher, “Sur les puissances de fonctions caracteristiques,”Comptes Rendus 246(1958), 694–696.
31. H. Teicher, “On the mixture of distributions,”Ann. Math. Stat. 31(1960), 55–73.
32. H. Teicher, “Identifiability of mixtures,”Ann. Math. Stat. 32(1961), 244–248.
33. E. C. TitchmarshThe Theory of FunctionsOxford Univ. Press, 1932; 2nd ed., 1939.