Acta Mathematica Hungarica

, Volume 151, Issue 2, pp 510–530 | Cite as

Almost sure limit behavior of Cesàro sums with small order

  • A. Gut
  • U. Stadtmüller


Various methods of summation for divergent series have been extended to analogs for sums of i.i.d. random variables. The present paper deals with a special class of matrix weighted sums of i.i.d. random variables where the weights \({a_{n,k}}\) are defined as the weights from Cesàro summability, i.e., \({a_{n,k}=\binom{n-k+\alpha-1}{n-k}/\binom{n+\alpha}{n}}\), where \({\alpha > 0}\). A strong law of large numbers (SLLN) has been shown to hold in this setting iff \({E {|X|}^{1/\alpha}<\infty}\), but a law of the iterated logarithm (LIL) has been shown for the case \({\alpha \geqq 1}\) only. We will study the case \({0 < \alpha < 1}\) in more detail, giving an LIL for \({1/2 < \alpha < 1}\) and some additional strong limit theorems under appropriate moment conditions for \({1/2 \leqq \alpha < 1}\).

Key words and phrases

Cesàro summation strong law law of the iterated logarithm exponential bound 

Mathematics Subject Classification

primary 60F15 60G50 secondary 40G05 


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  1. 1.
    Adler A., Rosalsky A.: On the strong law of large numbers for normed weighted sums of i.i.d. random variables, Stochastic Anal. Appl. 5, 467–483 (1987)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bingham N.H.: Variants on the law of the iterated logarithm, Bull. London Math. Soc. 18, 433–467 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    N. H. Bingham, Ch. Goldie and J. Teugels, Regular Variation, Cambridge University Press (Cambridge, 1987).Google Scholar
  4. 4.
    Chow Y. S., Lai T. L.: Limiting behavior of weighted sums of independent random variables, Ann. Probab. 1, 810–824 (1973)CrossRefzbMATHGoogle Scholar
  5. 5.
    M. Csörgő and P. Révész, Strong Approximations in Probability and Statistics, Academic Press (New York, 1981).Google Scholar
  6. 6.
    Déniel Y., Derriennic Y.: Sur la convergence presque sure, au sens de Cesàro d’ordre \({\alpha}\), \({0 < \alpha < 1}\), de variables aléatoires indépendantes et identiquement distribuées, Probab. Th. Rel. Fields 79, 629–636 (1988)CrossRefGoogle Scholar
  7. 7.
    Gut A.: Complete convergence for arrays, Period. Math. Hungar. 25, 51–75 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gut A.: Complete convergence and Cesàro summation for i.i.d. random variables, Probab. Th. Rel. Fields 97, 169–178 (1993)CrossRefzbMATHGoogle Scholar
  9. 9.
    A. Gut, Probability: A Graduate Course, 2nd edn. Springer-Verlag (New York, 2013).Google Scholar
  10. 10.
    A. Gut and U. Stadtmüller, Complete convergence of arrays of random variables in the vicinity of the LSL. Preprint, Ulm and Uppsala (2016a) (submitted).Google Scholar
  11. 11.
    Gut A., Stadtmüller U.: Strong laws for sequences in the vicinity of the LIL, Statistics & Probab. Letters 122, 63–72 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hartman P., Wintner A.: On the law of the iterated logarithm, Amer. J. Math. 63, 169–176 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Marcinkiewicz J., Zygmund A.: Sur les fonctions indépendantes, Fund. Math. 29, 60–90 (1937)zbMATHGoogle Scholar
  14. 14.
    Stadtmüller U.: A note on the law of iterated logarithm for weighted sums of random variables, Ann. Prob. 12, 35–44 (1984)CrossRefzbMATHGoogle Scholar
  15. 15.
    Strassen V.: A converse to the law of the iterated logarithm, Z. Wahrsch. verw. Gebiete 4, 265–268 (1966)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Department of Number Theory and ProbabilityUlm UniversityUlmGermany

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