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The Darling-Erdős theorem for sums of I.I.D. random variables
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  • Published: June 1989

The Darling-Erdős theorem for sums of I.I.D. random variables

  • Uwe Einmahl1 nAff2 

Probability Theory and Related Fields volume 82, pages 241–257 (1989)Cite this article

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  • 36 Citations

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Summary

We obtain a general Darling-Erdős type theorem for the maximum of appropriately normalized sums of i.i.d. mean zero r.v.'s with finite variances. We infer that the Darling-Erdős theorem holds in its classical formulation if and only ifE[X 2 1 {|X|≧t}]=o((loglogt)-1) ast→∞. Our method is based on an extension of the truncation techniques of Feller (1946) to non-symmetric r.v.'s. As a by-product we are able to reprove fundamental results of Feller (1946) dealing with lower and upper classes in the Hartman-Wintner LIL.

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References

  1. Csörgő, M., Révész, P.: Strong approximations in probability and statistics. New York: Academic Press 1981

    Google Scholar 

  2. Darling, D., Erdős, P.: A limit theorem for the maximum of normalized sums of independent random variables. Duke Math. J.23, 143–155 (1956)

    Google Scholar 

  3. Einmahl, U.: Strong invariance principles for partial sums of independent random vectors. Ann. Probab.15, 1419–1440 (1987)

    Google Scholar 

  4. Feller, W.: The law of the iterated logarithm for identically distributed random variables. Ann. Math.47, 631–638 (1946)

    Google Scholar 

  5. Itô, K., McKean, H.P., Jr.: Diffusion processes and their sample pathes. Berlin Heidelberg New York: Springer 1965

    Google Scholar 

  6. Jain, N., Jogdeo, K., Stout, W.: Upper and lower functions for martingales and mixing processes. Ann. Probab.3, 119–145 (1975)

    Google Scholar 

  7. Major, P.: An improvement of Strassen's invariance principle. Ann. Probab.7, 55–61 (1979)

    Google Scholar 

  8. Mason, D.: Personal communication (1987)

  9. Oodaira, H.: Some limit theorems for the maximum of normalized sums of weakly dependent random variables. In: Proceedings of the Third Japan-USSR Symposium on Probability Theory (eds., G. Maruyama and J.V. Prokhorov). Lect. Notes Math.550, 467–474. Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  10. Robbins, H., Siegmund, D.: Boundary crossing probabilities for the Wiener process and sample sums. Ann. Math. Stat.41, 1410–1429 (1970)

    Google Scholar 

  11. Shorack, G.R.: Extension of the Darling-Erdős theorem on the maximum of normalized sums. Ann. Probab.7, 1092–1096 (1979)

    Google Scholar 

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Author information

Author notes
  1. Uwe Einmahl

    Present address: Department of Mathematics, Indiana University, 47405, Bloomington, IN, USA

Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, 48824-1027, East Lansing, MI, USA

    Uwe Einmahl

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  1. Uwe Einmahl
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Cite this article

Einmahl, U. The Darling-Erdős theorem for sums of I.I.D. random variables. Probab. Th. Rel. Fields 82, 241–257 (1989). https://doi.org/10.1007/BF00354762

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  • Received: 13 March 1988

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00354762

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Classical Formulation
  • Type Theorem
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