Theorem 8

Let w(x) be a proper W-function. If, for a sequence of constants a_n > 0 and b_n, $n = 1,2,3,\ldots$,

$$n \left[ 1 - F (a_n x + b_n) \right] \Rightarrow w(x),$$

then w(x) is of the same type as one of the following W-functions:

$$\begin{array} {r@{~}c@{~}l} W_1 (a,x) & = & \begin{cases} \i... ...{for } x \gt 0, \end{cases} \\ W_3 (x) & = & e^{-x},\end{array}$$

where a is a positive constant.

Proof: The theorem follows immediately from Theorems 7 and 4.