Skip to main content

Almost certain behavior of row sums of double arrays

Part of the Lecture Notes in Mathematics book series (LNM,volume 861)

Abstract

The almost certain (probability one) behavior of row sums of double arrays of rowwise independent random variables Xn, i is investigated primarily when the Xn, i assume only the values zero and one. If knpn, the expected number of successes, grows more rapidly than C log n, then converges almost certainly to one. However, when the expected number of successes grows less rapidly than C log n, it no longer determines the asymptotic behavior of and the outcome depends upon additional stochastic assumptions. Two alternative sets of stochastic constraints are imposed and the behavior of under each is analyzed and contrasted. Finally, the case of exponential random variables is considered briefly.

Keywords

  • Triangular Array
  • Independent Column
  • Poisson Probability
  • Standard Normal Density
  • Poisson Case

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research supported by the National Science Foundation under Grant NSF-MCS-8005481.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, T. W. and Samuels, S. M. Some inequalities among Binomial and Poisson probabilities. Proc. Fifth Berkeley Symposium. Univ. of California Press, Vol. 1, 1–12.

    Google Scholar 

  2. Baxter, G., An analogue of the Law of the Iterated Logarithm. Proc. Amer. Math. Soc. 6 (1955), 177–181.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Chow, Y. S. and Teicher, H., Probability Theory: Independence, Interchange-ability, Martingales. Springer-Verlag, New York 1978.

    CrossRef  MATH  Google Scholar 

  4. Cramér, H., Su un teorema relativo alla legge uniforme dei grandi numeri. Giornale dell’ Istituto Italiano degli Attuari 5, #1 (1934), 1–13.

    MATH  Google Scholar 

  5. Hoeffding, W., On the distribution of the number of successes in independent trials. Ann. Math. Stat. 27 (1963), 713–721.

    CrossRef  MathSciNet  Google Scholar 

  6. Rosalsky, A. and Teicher, H., A limit theorem for double arrays. Ann. Prob. 9 (1981).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1981 Springer-Verlag

About this paper

Cite this paper

Teicher, H. (1981). Almost certain behavior of row sums of double arrays. In: Dugué, D., Lukacs, E., Rohatgi, V.K. (eds) Analytical Methods in Probability Theory. Lecture Notes in Mathematics, vol 861. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097322

Download citation

  • DOI: https://doi.org/10.1007/BFb0097322

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10823-8

  • Online ISBN: 978-3-540-36785-7

  • eBook Packages: Springer Book Archive