Generalized k-matches

• Published in: Statistics & probability letters Vol. 38; no. 2; pp. 167 - 175
• Main Authors: Herzog, Jonathan, McLaren, Christopher, Godbole, Anant P.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.1998; Elsevier
• Series: Statistics & Probability Letters
• 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; Coupling; k-matches; Stein-Chen method; Poisson approximation; Distribution function; Poisson distribution
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/S0167-7152(97)00169-7

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.