Abstract
Central limit theorems have played a paramount role in probability theory starting—in the case of independent random variables—with the DeMoivre- Laplace 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.
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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. Ahad. Nauk. SSSR 24 (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. Chung, A Course in Probability Theory, Harcourt Brace, New York, 1968; 2nd ed., Academic Press, New York, 1974.
W. Doeblin, “Sur deux problèmes de M. Kolmogorov concernant les chaines denom-brables,” Bull, Soc. Math. France 66 (1938), 210–220.
J. L. Doob, Stochastic Processes, Wiley 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 Wahrscheinlichkeitsrechnung,” Math, Zeit. 40 (1935), 521–559.
N. Friedman, M. Katz, and L. Koopmans, “Convergence rates for the central limit theorem,” Proc. Nat. Acad. Sei. 56 (1966), 1062–1065.
P. Hall and C. C. Heyde, Martingale Limit Theory and its Application, Academic Press, New York, 1980.
P. Lévy, Théorie de l’ addition des variables aleatoiries, Gauthier-Villars, Paris, 1937; 2nd ed., 1954.
J. Lindeberg, “Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung,” Math. Zeit. 15 (1922), 211–225.
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. Kozl. 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 variables, Acta Math. Acad. Sei. 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 Pompilj, pp. 141–148, Gubbio, 1971.
H. Teicher, “A classical limit theorem without invariance or reflection, Ann. 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.
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© 1988 Springer-Verlag New York Inc.
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Chow, Y.S., Teicher, H. (1988). Central Limit Theorems. In: Probability Theory. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0504-0_9
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