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Asymptotic expansions for large deviation probabilities in the strong law of large numbers
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  • Published: June 1989

Asymptotic expansions for large deviation probabilities in the strong law of large numbers

  • James Allen Fill1 

Probability Theory and Related Fields volume 81, pages 213–233 (1989)Cite this article

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Summary

LetX 1,X 2,... be a sequence of independent random variables with common distribution functionF having zero mean, and let (S n ) be the random walk of partial sums. The weak and strong laws of large numbers, respectively, imply that for any α∈ℝ and ε>0 the probabilitiesP{S m >α+εm} and

$$Pm: = P\{ S_n > \alpha + \varepsilon n {\text{for some }}n \geqq m\} $$

tend to 0 asm tends to ∞. Building upon work of Bahadur and Ranga Rao [Ann. Math. Stat.31, 1015–1027 (1960)], Siegmund [Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 107–113 (1975)], and Fill and Wichura [Probab. Th. Rel. Fields81, 189–212 (1989)], we produce complete asymptotic expansions for the probabilitiesP{S m >α+εm} andp m .

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References

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Authors and Affiliations

  1. Department of Mathematical Sciences, The John Hopkins University, 220 Maryland Hall, 34th and Charles Streets, 21218, Baltimore, MD, USA

    James Allen Fill

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  1. James Allen Fill
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Additional information

Written while the author was on leave from Stanford University at the University of Chicago

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Cite this article

Fill, J.A. Asymptotic expansions for large deviation probabilities in the strong law of large numbers. Probab. Th. Rel. Fields 81, 213–233 (1989). https://doi.org/10.1007/BF00319551

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  • Received: 20 August 1987

  • Revised: 05 August 1988

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00319551

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Keywords

  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Asymptotic Expansion
  • Mathematical Biology
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