Rocky Mountain Journal of Mathematics

On a Stationary, Triple-Wise Independent, Absolutely Regular Counterexample to the Central Limit Theorem

Richard C. Bradley

Full-text: Open access

Article information

Source
Rocky Mountain J. Math., Volume 37, Number 1 (2007), 25-44.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1181069318

Digital Object Identifier
doi:10.1216/rmjm/1181069318

Mathematical Reviews number (MathSciNet)
MR2316436

Zentralblatt MATH identifier
1136.60015

Subjects
Primary: 60G10: Stationary processes 60F05: Central limit and other weak theorems

Keywords
Strictly stationary triple-wise independent central limit theorem

Citation

Bradley, Richard C. On a Stationary, Triple-Wise Independent, Absolutely Regular Counterexample to the Central Limit Theorem. Rocky Mountain J. Math. 37 (2007), no. 1, 25--44. doi:10.1216/rmjm/1181069318. https://projecteuclid.org/euclid.rmjm/1181069318


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References

  • P. Billingsley, Probability and measure, 3rd ed., Wiley, New York, 1995.
  • R.C. Bradley, A stationary, pairwise independent, absolutely regular sequence for which the central limit theorem fails, Probab. Theory Related Fields 81 (1989), 1-10.
  • --------, Introduction to strong mixing conditions, Vol. 1, Technical Report, Dept. of Mathematics, Indiana University, Bloomington, Custom Publ. of I.U., Bloomington, 2005.
  • J.A. Cuesta and C. Matrán, On the asymptotic behavior of sums of pairwise independent random variables, Statist. Probab. Lett. 11 (1991), 201-210 (Correction Ibid. 12 (1991), 183).
  • Yu.A. Davydov, Mixing conditions for Markov chains, Theory Probab. Appl. 18 (1973), 312-328.
  • M. Denker, Uniform integrability and the central limit theorem for strongly mixing processes, in Dependence in probability and statistics (E. Eberlein and M.S. Taqqu, eds.), Birkhäuser, Boston, 1986.
  • N. Etemadi, An elementary proof of the strong law of large numbers, Z. Wahrsch. Verw. Gebeite 55 (1981), 119-122.
  • N. Herrndorf, Stationary strongly mixing sequences not satisfying the central limit theorem, Ann. Probab. 11 (1983), 809-813.
  • S. Janson, Some pairwise independent sequences for which the central limit theorem fails, Stochastics 23 (1988), 439-448.
  • F. Merlevède and M. Peligrad, The functional central limit theorem under the strong mixing condition, Ann. Probab. 28 (2000), 1336-1352.
  • T. Mori and K. Yoshihara, A note on the central limit theorem for stationary strong-mixing sequences, Yokohama Math J. 34 (1986), 143-146.
  • A.R. Pruss, A bounded $N$-tuplewise independent and identically distributed counterexample to the CLT, Probab. Theory Related Fields 111 (1998), 323-332.
  • M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci. USA 42 (1956), 43-47.
  • V.A. Volkonskii and Yu.A. Rozanov, Some limit theorems for random functions I, Theory Probab. Appl. 4 (1959), 178-197.