On Location and Scale Functions of a Class of Limiting Processes with Application to Extreme Value Theory

Abstract

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.