Generalized k-matches

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 wa...

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Bibliographic Details
Published in:Statistics & probability letters 1998-06, Vol.38 (2), p.167-175
Main Authors: Herzog, Jonathan, McLaren, Christopher, Godbole, Anant P.
Format: Article
Language:English
Subjects:
Coupling
Distribution theory
Exact sciences and technology
k-matches
k-matches Poisson approximation Coupling Stein-Chen method
Mathematics
Nonparametric inference
Poisson approximation
Probability and statistics
Sciences and techniques of general use
Statistics
Stein-Chen method
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Summary: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.
ISSN:0167-7152
1879-2103
DOI:10.1016/S0167-7152(97)00169-7