Summary
A strictly stationary finite-state non-degenerate random sequence is constructed which satisfies pairwise independence and absolute regularity but fails to satisfy a central limit theorem. The mixing rate for absolute regularity is only slightly slower than that in a corresponding central limit theorem of Ibragimov.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bradley, R.C.: Counterexamples to the central limit theorem under strong mixing conditions. In: Revesz, P. (ed.) Colloquia mathematica societatis Janos Bolyai 36. Limit theorems in probability and statistics, Veszprem (Hungary), 1982, pp. 153–172. Amsterdam: North-Holland, 1985
Bradley, R.C.: On the central limit question under absolute regularity. Ann. Probab. 13, 1314–1325 (1985)
Bradley, R.C.: On some results of M.I. Gordin: a clarification of a misunderstanding. J. Theor. Probab. 1, 115–119 (1988)
Davydov, Yu.A.: Mixing conditions for Markov chains. Theory Probab. Appl. 18, 312–328 (1973)
Etemadi, N.: An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheor. Verw. Geb. 55, 119–122 (1981)
Herrndorf, N.: Stationary strongly mixing sequences not satisfying the central limit theorem. Ann. Probab. 11, 809–813 (1983)
Ibragimov, I.A., Linnik, Yu.V.: Independent and stationary sequences of random variables. Groningen: Wolters-Noordhoff, 1971
Janson, S.: Some pairwise independent sequences for which the central limit theorem fails. Stochastics 23, 439–448 (1988)
Rogozin, B.A.: An estimate of the remainder term in limit theorems of renewal theory. Theory Probab. Appl. 18, 662–677 (1973)
Volkonskii, V.A., Rozanov, Yu.A.: Some limit theorems for random functions I. Theory Probab. Appl. 4, 178–197 (1959)
Volkonskii, V.A., Rozanov, Yu.A.: Some limit theorems for random functions II. Theory Probab. Appl. 6, 186–198 (1961)
Author information
Authors and Affiliations
Additional information
This work was partially supported by NSF grant DMS 86-00399
Rights and permissions
About this article
Cite this article
Bradley, R.C. A stationary, pairwise independent, absolutely regular sequence for which the central limit theorem fails. Probab. Th. Rel. Fields 81, 1–10 (1989). https://doi.org/10.1007/BF00343735
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00343735