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A remark on the central limit question for dependent random variables

Published online by Cambridge University Press:  14 July 2016

Richard C. Bradley Jr
Richard C. Bradley Jr*
Affiliation:
Columbia University
*
∗Postal address: Department of Mathematical Statistics, Columbia University, New York, NY 10027, U.S.A.
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Abstract

Given a strictly stationary sequence {Xk, k = …, −1,0,1, …} of r.v.'s one defines for n = 1, 2, 3 …, . Here an example of {Xk} is given with finite second moments, for which Var(X1 + … + Xn)→∞ and ρ n → 0 as n→∞, but (X1 + … + Xn) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.

Keywords

STRICTLY STATIONARYMIXING CONDITIONMAXIMAL CORRELATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 94 - 101
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Research partially supported by the National Science Foundation and the Office of Naval Research.

References

Davydov, Yu. A. (1969) On the strong mixing property for Markov chains with a countable number of states. Soviet Math. 10, 825–827.Google Scholar
Gebelein, H. (1941) Das statistische Problem der Korrelation als Variations- und Eigenwertproblem und sein Zusammenhang mit der Ausgleichungsrechnung. Z. Angew. Math. Mech. 21, 364–379.Google Scholar
Helson, H. and Sarason, D. (1967) Past and future. Math. Scand. 21, 5–16.Google Scholar
Helson, H. and Szego, G. (1960) A problem in prediction theory. Ann. Mat. Pura Appl. (4) 51, 107–138.Google Scholar
Hirschfeld, O. (1935) A connection between correlation and contingency. Proc. Camb. Phil. Soc. 31, 520–524.Google Scholar
Ibragimov, I. A. (1959) Some limit theorems for strict-sense stationary stochastic processes (in Russian). Dokl. Akad. Nauk. SSSR 125, 711–714.Google Scholar
Ibragimov, I. A. (1975) A note on the central limit theorem for dependent random variables. Theory Prob. Appl. 20, 135–141.Google Scholar
Ibragimov, I. A. and Linnik, Yu. V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Kolmogorov, A. N. and Rozanov, Yu. A. (1960) On strong mixing conditions for stationary Gaussian processes. Theory Prob. Appl. 5, 204–208.CrossRefGoogle Scholar
Rosenblatt, M. (1956) A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA 42, 43–47.Google Scholar
Sarason, D. (1972) An addendum to ‘Past and future’. Math. Scand. 30, 62–64.Google Scholar
Witsenhausen, H. S. (1975) On sequences of pairs of dependent random variables. SIAM J. Appl. Math. 28, 100–113.CrossRefGoogle Scholar