Abstract
Central limit theorems have played a paramount role in probability theory starting—in the case of independent random variables—with the DeMoivreLaplace version and culminating with that of Lindeberg-Feller. The term “central” refers to the pervasive, although nonunique, role of the normal distribution as a limit of d.f.s of normalized sums of (classically independent) random variables. Central limit theorems also govern various classes of dependent random variables and the cases of martingales and interchangeable random variables will be considered.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Anscombe, “ Large sample theory of sequential estimation,”Proc. Cambr. Philos. Soc. 48 (1952) 600–607.
S. Bernstein, “ Several comments concerning the limit theorem of Liapounov,”Dokl. Akad. Nauk. SSSR24 (1939), 3–7.
A. C. Berry, “ The accuracy of the Gaussian approximation to the sum of independent variates,”Trans. Amer. Math. Soc.49 (1941), 122–136.
J. Blum, D. Hanson, and J. Rosenblatt, “On the CLT for the sum of a random number of random variables,”Z. Wahr. Verw. Geb.1 (1962–1963), 389–393.
J. Blum, H. Chernoff, M. Rosenblatt, and H. Teicher, “Central limit theorems for interchangeable processes,”Can. Jour. Math.10 (1958), 222–229.
K. L. ChungA Course in Probability TheoryHarcourt Brace, New York, 1968; 2nd ed., Academic Press, New York, 1974.
W. Doeblin, “ Sur deux problèmes de M. Kolmogorov concernant les chaînes denombrables,”Bull Soc. Math. France66 (1938), 210–220.
J. L. DoobStochastic ProcessesWiley, New York, 1953.
A. Dvoretzky, “Asymptotic normality for sums of dependent random variables,”Proc. Sixth Berkeley Symp. on Stat. and Prob.1970, 513–535.
P. Erdos and M. Kac, “On certain limit theorems of the theory of probability,”Bull. Amer. Math. Soc.52 (1946), 292–302.
C. Esseen, “Fourier analysis of distribution functions,”Acta Math.77 (1945), 1–125.
W. Feller, “ Über den Zentralen Grenzwertsatz der ahrscheinlichkeitsrechnung,” MathZeit.40 (1935), 521–559.
N. Friedman, M. Katz, and L. Koopmans, “Convergence rates for the central limit theorem,”Proc. Nat. Acad. Sci.56 (1966), 1062–1065.
P. Hall and C. C. HeydeMartingale Limit Theory and its ApplicationAcademic Press, New York, 1980.
K. KnoppTheory and Application of Infinite SeriesStechert-Hafner, New York, 1928.
P. LévyThéorie de l’addition des variables aléatoiriesGauthier-Villars, Paris, 1937; 2nd ed., 1954.
J. Lindeberg, “Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung,”Math. Zeit.15 (1922), 211–225.
D. L. McLeish, “Dependent Central Limit Theorems and invariance principles,”Ann. Prob.2 (1974), 620–628.
J. Mogyorodi, “A CLT for the sum of a random number of independent random variables,”Magyor. Tud. Akad. Mat. Kutato Int. Közl.7 (1962), 409–424.
A. Renyi, “Three new proofs and a generalization of a theorem of Irving Weiss,”Magyor. Tud. Akad. Mat. Kutato Int. Közl.7 (1962), 203–214.
A. Renyi, “On the CLT for the sum of a random number of independent random variablesActa Math. Acad. Sci. Hung.11 (1960), 97–102.
B. Rosen, “On the asymptotic distribution of sums of independent, identically distributed random variables,”Arkiv for Mat.4 (1962), 323–332.
D. Siegmund, “On the asymptotic normality of one-sided stopping rules,”Ann. Math. Stat.39 (1968), 1493–1497.
H. Teicher, “On interchangeable random variables,”Studi di Probabilita Statistica e Ricerca Operativa in Onore di Giuseppe Pompiljpp. 141–148, Gubbio, 1971.
H. Teicher, “A classical limit theorem without invariance or reflectionAnn. Math. Stat.43 (1973), 702–704.
P. Van Beek, “An application of the Fourier method to the problem of sharpening the Berry-Esseen inequality,”Z. Wahr.23 (1972), 187–197.
I. Weiss, “Limit distributions in some occupancy problems,”Ann. Math. Stat.29 (1958), 878–884.
V. Zolotarev, “An absolute estimate of the remainder term in the C.L.T.,”Theor. Prob. and its Appl.11 (1966), 95–105.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
Chow, Y.S., Teicher, H. (1997). Central Limit Theorems. In: Probability Theory. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1950-7_9
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1950-7_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-40607-7
Online ISBN: 978-1-4612-1950-7
eBook Packages: Springer Book Archive