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Waiting Time Distribution for the Emergence of Superpatterns

Author

Listed:
  • Anant P. Godbole

    (East Tennessee State University)

  • Martha Liendo

    (East Tennessee State University)

Abstract

Consider a sequence { X n } n = 1 ∞ $\{X_{n}\}_{n=1}^{\infty }$ of i.i.d. uniform random variables taking values in the alphabet set {1, 2,…, d}. A k-superpattern is a realization of { X n } n = 1 t $\{X_{n}\}_{n=1}^{t}$ that contains, as an embedded subsequence, each of the non-order-isomorphic subpatterns of length k. We focus on the (non-trivial) case of d = k = 3 and study the waiting time distribution of τ = inf { t ≥ 1 : { X n } n = 1 t is a superpattern } $\tau =\inf \{t\ge 1:\{X_{n}\}_{n=1}^{t}\ \text {is\ a\ superpattern}\}$ . Our restricted set-up leads to proofs that are very combinatorial in nature, since we are essentially conducting a string analysis.

Suggested Citation

  • Anant P. Godbole & Martha Liendo, 2016. "Waiting Time Distribution for the Emergence of Superpatterns," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 517-528, June.
    • Handle: RePEc:spr:metcap:v:18:y:2016:i:2:d:10.1007_s11009-015-9439-6
    DOI: 10.1007/s11009-015-9439-6
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    References listed on IDEAS

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    1. Sunil Abraham & Greg Brockman & Stephanie Sapp & Anant P. Godbole, 2013. "Omnibus Sequences, Coupon Collection, and Missing Word Counts," Methodology and Computing in Applied Probability, Springer, vol. 15(2), pages 363-378, June.
    2. Graham J. G. Upton, 1994. "Picturing the 1992 British General Election," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 157(2), pages 231-252, March.
    3. David E. Altig & Jagadeesh Gokhale, 1994. "Health care reform from a generational perspective," Economic Commentary, Federal Reserve Bank of Cleveland, issue Apr.
    4. James C. Fu, 2012. "On the Distribution of the Number of Occurrences of an Order-Preserving Pattern of Length Three in a Random Permutation," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 831-842, September.
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