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Generalized k-matches

Author

Listed:
  • Herzog, Jonathan
  • McLaren, Christopher
  • Godbole, Anant P.

Abstract

Suppose that an urn contains m distinguishable balls, and that these balls are sampled (with replacement), thus generating a sequence of colors. Many questions can be asked about this sequence; the distribution of the time until a color is sampled twice within a memory window of size k (i.e., the waiting time till the first k-match) was derived by Arnold (1972). Next, Burghardt et al. (1994) proved that the limiting distribution of the number of k-matches in the first n draws is Poisson if k = o(m). An even more general question is discussed here: if, for every draw from the urn, a random k-sample is taken of the previous draws, what is the distribution of the number of generalized k-matches? Our solution resolves a question of Glen Meeden (see Arnold, 1972). Extensions to the case where the k-sample is drawn from the (union of the) past and the future are provided, and the case of non-uniform selection probabilities is treated.

Suggested Citation

  • Herzog, Jonathan & McLaren, Christopher & Godbole, Anant P., 1998. "Generalized k-matches," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 167-175, June.
    • Handle: RePEc:eee:stapro:v:38:y:1998:i:2:p:167-175
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    References listed on IDEAS

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    1. Burghardt, Patrick D. & Godbole, Anant P. & Prengaman, Amy B., 1994. "A Poisson approximation for the number of k-matches," Statistics & Probability Letters, Elsevier, vol. 21(1), pages 1-8, September.
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