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A note on the number of records near the maximum

✍ Scribed by Yun Li

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
83 KB
Volume
43
Category
Article
ISSN
0167-7152

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✦ Synopsis

Let {Xn; nΒΏ1} be a sequence of independent identically distributed random variables with the continuous distribution function F(x). Let Kn(a) denote the number of values j ∈ {1; 2; : : : ; n} for which Xj ∈ (Mn -a; Mn], where Mn = max{X1; : : : ; Xn} and a is a positive constant. In this paper we prove that limnβ†’βˆž E(Kn(a)) = 1 if and only if Kn(a) converges in probability to one, if and only if when F(x) has a thick tail. Furthermore, we will give a necessary and su cient condition for Kn(a)

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In this paper, we prove that XT(G) = 5 for any Halin graph G with A(G) = 4, where A(G) and XT(G) denote the maximal degree and the total chromatic number of G, respectively.