Waiting Time Distribution for the Emergence of Superpatterns

Anant P. Godbole () and Martha Liendo
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Anant P. Godbole: East Tennessee State University
Martha Liendo: East Tennessee State University

Methodology and Computing in Applied Probability, 2016, vol. 18, issue 2, 517-528

Abstract: 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.

Keywords: Superpattern; Waiting time distribution; Generating function; 60C05; 05A15 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s11009-015-9439-6

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