Theorem 7

Let $\phi(x)$ and w(x) be, respectively, a proper frequency distribution and a proper W-function. Suppose further that $\phi(x) = e^{-w(x)}$. For any sequence of constants a_n > 0 and b_n, $n = 1,2,3,\ldots$,

$$n \left[ 1 - F (a_n x + b_n) \right] \Rightarrow w(x), \quad... ...xt{if and only if} \quad F^n (a_n x + b_n) \Rightarrow \phi(x).$$

Proof: This is a restatement of Corollary 5 with fewer restrictions on the convergence of $n \left[ 1 - F (a_n x + b_n ) \right]$.

Suppose first that there exists a proper frequency distribution $\phi(x)$ such that

$$F^n (a_n x + b_n ) \Rightarrow \phi (x).$$

Then, by Corollary 5,

$$\lim_{n\to\infty} n \left[ 1 - F (a_n x + b_n ) \right] = - \log \phi (x)$$

for all x. Hence, certainly,

$$n \left[ 1 - F (a_n x + b_n)\right] \Rightarrow - \log \phi (x) = w(x).$$

Now, assume that there exists a proper W-function w(x), such that

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

Let x₀ be any continuity point of w(x) or any point at which w(x) is infinite. Now, by Theorem 5,

$$\lim_{n\to\infty} F^n (a_n x_0 + b_n) = e^{-w(x_0)} = \phi (x_0).$$

But, by inspection of the relationship between $\phi(x)$ and w(x), if x₀ is either a continuity point of w(x) or a point at which w(x) is infinite, then $\phi(x)$ is continuous at x = x₀. Also, if $\phi(x)$ is continuous at x = x₀, then w(x₀) is infinite or w(x) is continuous at x = x₀. Hence, for every continuity point of $\phi(x)$,

$$\lim_{n\to\infty} F^n (a_n x + b_n) = e^{-w(x)} = \phi (x)$$

and so

$$F^n (a_n x + b_n) \Rightarrow e^{-w(x)} = \phi (x).$$

From the characteristics of a proper W-function (Definition 3), it follows that $\phi(x)$ is a proper frequency distribution.