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Distribution of the number of consecutive records
Author(s) -
Chern HuaHuai,
Hwang HsienKuei,
Yeh YeongNan
Publication year - 2000
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/1098-2418(200010/12)17:3/4<169::aid-rsa1>3.0.co;2-k
Subject(s) - mathematics , bijection, injection and surjection , independent and identically distributed random variables , poisson distribution , hypergeometric distribution , combinatorics , random variable , sequence (biology) , asymptotic distribution , distribution (mathematics) , permutation (music) , discrete mathematics , bijection , statistics , mathematical analysis , physics , estimator , biology , genetics , acoustics
We study the distribution of the number ξ n , r of times that r consecutive records occur in a sequence of n independent and identically distributed random variables from a common continuous distribution, or equivalently, in a random permutation of n elements. We show that the asymptotic distribution of ξ n , r exists and is Poisson for r =1, 2 and non‐Poisson for r ≥3. Precise asymptotic results are derived for four probability distances of the associated approximations: Fortet‐Mourier, total variation, Kolmogorov, and point metric. In particular, the distributions of ξ n , r have the specific property that the last three distances are asymptotically of the same behavior for r ≥2. We also provide interesting combinatorial bijections for ξ n , 2 and compute explicitly the limiting law for ξ n , 3 in terms of Kummer's confluent hypergeometric functions. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 169–196 2000