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On the Number of Appearances of a Word in a Sequence of I.I.D. Trials

Author

Listed:
  • Ourania Chryssaphinou

    (University of Athens)

  • Stavros Papastavridis

    (University of Athens)

  • Eutichia Vaggelatou

    (University of Athens)

Abstract

Let X 1,...,X n be a sequence of i.i.d. random variables taking values in an alphabet Ω=ω1,...,ωq,q ≥ 2, with probabilities P(X a=ωi)=p i,a=1,...,n,i=1,...,q. We consider a fixed h-letter word W=w1...wh which is produced under the above scheme. We define by R(W) the number of appearances of W as Renewal (which is equal with the maximum number of non-overlapping appearances) and by N(W) the number of total appearances of W (overlapping ones) in the sequence X a 1≤ a1≤n under the i.i.d. hypothesis. We derive a bound on the total variation distance between the distribution ℒ(R(W)) of the r.v. R(W) and that of a Poisson with parameter E(R(W)). We use the Stein-Chen method and related results from Barbour et al. (1992), as well as, combinatorial results from Schbath (1995b) concerning the periodic structure of the word W. Analogous results are obtained for the total variation distance between the distribution of the r.v. N(W) and that of an appropriate Compound Poisson r.v. Related limit theorems are obtained and via numerical computations our bounds are presented in tables.

Suggested Citation

  • Ourania Chryssaphinou & Stavros Papastavridis & Eutichia Vaggelatou, 1999. "On the Number of Appearances of a Word in a Sequence of I.I.D. Trials," Methodology and Computing in Applied Probability, Springer, vol. 1(3), pages 329-348, October.
    • Handle: RePEc:spr:metcap:v:1:y:1999:i:3:d:10.1023_a:1010042628865
    DOI: 10.1023/A:1010042628865
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    References listed on IDEAS

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    1. Gerber, Hans U. & Li, Shuo-Yen Robert, 1981. "The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 11(1), pages 101-108, March.
    2. Fousler, David E. & Karlin, Samuel, 1987. "Maximal success durations for a semi-Markov process," Stochastic Processes and their Applications, Elsevier, vol. 24(2), pages 203-224, May.
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