Almost Sure Convergence in Extreme Value Theory

✍ Scribed by Shihong Cheng; Liang Peng; Yongcheng Qi

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 263 KB
• Volume: 190
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19981900104

✦ Synopsis

Abstract

Let X~1~, …, X~n~ be independent random variables with common distribution function F. Define
equation image
and let G(x) be one of the extreme‐value distributions. Assume F ∈ D(G), i.e., there exist a~n~> 0 and b~n~ ∈ ℝ such that
equation image
.

Let l(−∞,x)(·) denote the indicator function of the set (−∞,x) and S(G) =: {x : 0 < G(x) < 1}. Obviously, 1(−∞,x)((M~n~−b~n~)/a~n~) does not converge almost surely for any x ∈ S(G). But we shall prove
equation image
.