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Journal of Theoretical Probability

, Volume 9, Issue 2, pp 413–428 | Cite as

Divergence rates for the number of rare numbers

  • Eric S. Key
Article

Abstract

Suppose thatX1,X2, ... is a sequence of i.i.d. random variables taking value inZ+. Consider the random sequenceA(X)≡(X1,X2,...). LetY n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofY n /E[Y n ] and establish conditions under which we have almost sure convergence to 1. We also find conditions under which we dtermine the rate of growth ofE[Y n ]. These results extend earlier work by the author.

Key Words

Rare numbers almost sure convergence subadditive 

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References

  1. 1.
    Cuevas, Antonio, and Walters, Gilbert G. (1992). A note on estimation of generalized densities.Comm. Statist. Theory Methods 21, 1807–1821.Google Scholar
  2. 2.
    Eisenberg, B., Stengle, G., and Strang, G. (1993). The asymptotic probability of a tie for first place.Ann. Appl. Prob. 3, 731–745.Google Scholar
  3. 3.
    Key, Eric S. (1992). Rare numbers.J. Theoret. Prob. 5, 375–389.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Eric S. Key
    • 1
  1. 1.Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukee

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