TY - JOUR
T1 - Almost sure convergence in extreme value theory
AU - Cheng, Shihong
AU - Peng, Liang
AU - Qi, Yongcheng
PY - 1998
Y1 - 1998
N2 - Let X1 , . . . , Xn be independent random variables with common distribution function F. Define Mn := max Xi1≤i≤n and let G(cursive Greek chi) be one of the extreme - value distributions. Assume F ∈ D(G), i.e., there exist an > 0 and bn ∈ double-struck R sign such that P{(Mn - bn)/an ≤ cursive Greek chi} → G(cursive Greek chi), for cursive Greek chi ∈ double-struck R sign . Let 1(-∞,cursive Greek chi](·) denote the indicator function of the set (-∞, cursive Greek chi] and S(G) =: {cursive Greek chi : 0 < G(cursive Greek chi) < 1}. Obviously, 1(-∞,cursive Greek chi]((Mn - bn)/an) does not converge almost surely for any cursive Greek chi ∈ S(G). But we shall prove (equation presented).
AB - Let X1 , . . . , Xn be independent random variables with common distribution function F. Define Mn := max Xi1≤i≤n and let G(cursive Greek chi) be one of the extreme - value distributions. Assume F ∈ D(G), i.e., there exist an > 0 and bn ∈ double-struck R sign such that P{(Mn - bn)/an ≤ cursive Greek chi} → G(cursive Greek chi), for cursive Greek chi ∈ double-struck R sign . Let 1(-∞,cursive Greek chi](·) denote the indicator function of the set (-∞, cursive Greek chi] and S(G) =: {cursive Greek chi : 0 < G(cursive Greek chi) < 1}. Obviously, 1(-∞,cursive Greek chi]((Mn - bn)/an) does not converge almost surely for any cursive Greek chi ∈ S(G). But we shall prove (equation presented).
KW - Almost sure convergence
KW - Arithmetic means
KW - Extreme value distribution
KW - Logarithmetic means
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U2 - 10.1002/mana.19981900104
DO - 10.1002/mana.19981900104
M3 - Article
AN - SCOPUS:0002038995
SN - 0025-584X
VL - 190
SP - 43
EP - 50
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
ER -