Let $Y_n (n = 0, 1, \cdots)$ be a random variable and suppose that for suitably chosen constants $a_n$ and $b_n (b_n > 0)$ and each $t \in (0, \infty)$, the random variable $b_n Y_{\lbrack nt \rbrack} + a_n$ has a limit distribution function $G_t$. If $G_1$ is non-degenerate there are only two ways in which $G_t$ is related to $G_1$: there exist real constants $c$ and $\theta$ such that either $G_t(x) = G_1(c + t^\theta(x - c))$ for all $t > 0$ or else $G_t(x) = G_1(x + c \log t)$ for all $t > 0$. This result provides a very short derivation of the three types of extreme value limit distributions.
Publié le : 1975-02-14
Classification:
Limiting process,
type,
location and scale functions,
extreme-value distribution functions,
60G99,
62E20,
62G30
@article{1176996460,
author = {Weissman, Ishay},
title = {On Location and Scale Functions of a Class of Limiting Processes with Application to Extreme Value Theory},
journal = {Ann. Probab.},
volume = {3},
number = {6},
year = {1975},
pages = { 178-181},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996460}
}
Weissman, Ishay. On Location and Scale Functions of a Class of Limiting Processes with Application to Extreme Value Theory. Ann. Probab., Tome 3 (1975) no. 6, pp. 178-181. http://gdmltest.u-ga.fr/item/1176996460/