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The convergence rate for the strong law of large numbers: General lattice distributions
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  • Published: June 1989

The convergence rate for the strong law of large numbers: General lattice distributions

  • James Allen Fill1 &
  • Michael J. Wichura2 

Probability Theory and Related Fields volume 81, pages 189–212 (1989)Cite this article

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  • 3 Citations

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Summary

LetX 1,X 2, ... be a sequence of independent random variables with common lattice distribution functionF having zero mean, and let (S n ) be the random walk of partial sums. The strong law of large numbers (SLLN) implies that for any α∈ℝ and ε>0

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

decreases to 0 asm increases to ∞. Under conditions on the moment generating function ofF, we obtain the convergence rate by determiningp m up to asymptotic equivalence. When α=0 and ε is a point in the lattice forF, the result is due to Siegmund [Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 107–113 (1975); but this restriction on ε precludes all small values of ε, and these values are the most interesting vis-à-vis the SLLN. Even when α=0 our result handles important distributionsF for which Siegmund's result is vacuous, for example, the two-point distributionF giving rise to simple symmetric random walk on the integers. We also identify for both lattice and non-lattice distributions the behavior of certain quantities in the asymptotic expression forp m as ε decreases to 0.

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References

  1. Bahadur, R.R., Rao, R. Ranga: On deviations of the sample mean. Ann. Math. Stat.31, 1015–1027 (1960)

    Google Scholar 

  2. Blackwell, d., Hodges, J.L., Jr.: The probability in the extreme tail of a convolution. Ann. Math. Stat.30, 1113–1120 (1959)

    Google Scholar 

  3. Chung, K.L.: A course in probability theory, 2nd ed. New York: Academic Press 1974

    Google Scholar 

  4. Daniels, H.E.: Tail probability approximations. Int. Stat. Rev.55, 37–48 (1987)

    Google Scholar 

  5. Feller, W.: An introduction to probability theory and its applications, Vol. 2, 2nd ed. New York: Wiley 1971

    Google Scholar 

  6. Fill, J.A.: convergence rates related to the strong law of large numbers. Stanford University Technical Report No. 14 (1980)

  7. Fill, J.A.: Convergence rates related to the strong law of large numbers. Ann. Probab.11, 123–143 (1983)

    Google Scholar 

  8. Fill, J.A.: Asymptotic expansions for large deviation probabilities in the strong law of large numbers. Probab. Th. Rel. Fields81, 213–233 (1989)

    Google Scholar 

  9. Lai, T.Z.: Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Probab.4, 51–66 (1976)

    Google Scholar 

  10. Lugannani, R., Rice, S.: Saddle point approximation for the distribution of the sum of independent random variables. Adv. Appl. Probab.12, 475–490 (1980)

    Google Scholar 

  11. Müller, D.W.: Verteilungs-Invarianzprinzipien für das Gesetz der großen Zahlen. Z. Wahrscheinlichkeitstheor. Verw. Geb.10, 173–192 (1968)

    Google Scholar 

  12. Siegmund, D.: Large deviation probabilities in the strong law of large numbers. Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 107–113 (1975)

    Google Scholar 

  13. Siegmund, D.: Sequential analysis. New York Berlin Heidelberg: Springer 1985

    Google Scholar 

  14. Spitzer, F.: A Tauberian theorem and its probability interpretation. Trans. Am. Math. Soc.94, 150–169 (1960)

    Google Scholar 

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Author information

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

  2. Department of Statistics, The University of Chicago, 5734 S. University Avenue, 60637, Chicago, IL, USA

    Michael J. Wichura

Authors
  1. James Allen Fill
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  2. Michael J. Wichura
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The work of this author was carried out in part while at Stanford University and in part while on leave at the University of Chicago

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Fill, J.A., Wichura, M.J. The convergence rate for the strong law of large numbers: General lattice distributions. Probab. Th. Rel. Fields 81, 189–212 (1989). https://doi.org/10.1007/BF00319550

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  • Received: 20 August 1987

  • Revised: 08 August 1988

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00319550

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Keywords

  • Generate Function
  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Convergence Rate
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