## Abstract

Let X_{1} , . . . , X_{n} be independent random variables with common distribution function F. Define M_{n} := max X_{i1≤i≤n} and let G(cursive Greek chi) be one of the extreme - value distributions. Assume F ∈ D(G), i.e., there exist a_{n} > 0 and b_{n} ∈ double-struck R sign such that P{(M_{n} - b_{n})/a_{n} ≤ 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}]((M_{n} - b_{n})/a_{n}) does not converge almost surely for any cursive Greek chi ∈ S(G). But we shall prove (equation presented).

Original language | English (US) |
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Pages (from-to) | 43-50 |

Number of pages | 8 |

Journal | Mathematische Nachrichten |

Volume | 190 |

DOIs | |

State | Published - 1998 |

## Keywords

- Almost sure convergence
- Arithmetic means
- Extreme value distribution
- Logarithmetic means