Almost Sure Convergence in Extreme Value Theory

Abstract

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 .