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Limit Theorems for Deviation Means of Independent and Identically Distributed Random Variables

Author

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  • Mátyás Barczy

    (University of Szeged)

  • Zsolt Páles

    (University of Debrecen)

Abstract

We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for so-called deviation means of independent and identically distributed random variables. (For the strong law of large numbers, we suppose only pairwise independence instead of (total) independence.) The class of deviation means is a special class of M-estimators or more generally extremum estimators, which are well studied in statistics. The assumptions of our limit theorems for deviation means seem to be new and weaker than the known ones for M-estimators in the literature. In particular, our results on the strong law of large numbers and on the central limit theorem generalize the corresponding ones for quasi-arithmetic means due to de Carvalho (Am Stat 70(3):270–274, 2016) and the ones for Bajraktarević means due to Barczy and Burai (Aequ Math 96(2):279–305, 2022).

Suggested Citation

  • Mátyás Barczy & Zsolt Páles, 2023. "Limit Theorems for Deviation Means of Independent and Identically Distributed Random Variables," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1626-1666, September.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:3:d:10.1007_s10959-022-01225-6
    DOI: 10.1007/s10959-022-01225-6
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    References listed on IDEAS

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    1. Rubin, Herman & Rukhin, Andrew L., 1983. "Convergence rates of large deviations probabilities for point estimators," Statistics & Probability Letters, Elsevier, vol. 1(4), pages 197-202, June.
    2. Miguel de Carvalho, 2016. "Mean, What do You Mean?," The American Statistician, Taylor & Francis Journals, vol. 70(3), pages 270-274, July.
    3. Miguel Arcones, 2006. "Large deviations for M-estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 21-52, March.
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