Corollary 11

Let E be the set of all points on the x axis. If for a sequence of constants a_n > 0 and b_n, $n = 1,2,3,\ldots$,

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

where w(x) is a proper W-function, then for any $x \in E$,

(a)

Fⁿ(a_nx + b_n) converges uniformly to e^-w(x), and

(b)

G_m,n (a_n x + b_n) converges uniformly to

$$e^{-w(x)} \sum^{m-1}_{k=0} \frac{[w(x)]^k}{k!}, \qquad (\text{for finite } m).$$

Proof: By Corollary 5, if

$$n [ 1 - F (a_n x + b_n)] \Rightarrow w(x), \quad \text{then} \quad F^n (a_n x + b_n) \Rightarrow \phi (x),$$

where $\phi(x)$ is a proper frequency distribution. But, by Corollary 4, $\phi(x)$ is continuous for every $x \in E$, and so

$$\lim_{n\to\infty} F^n (a_n x + b_n) = \phi (x), \qquad \text{for every } x \in E,$$

and $\phi(x)$ is a continuous frequency distribution. It follows that by Theorem 11, Fⁿ (a_n x + b_n) converges uniformly to $\phi(x)$.

By Theorem 9, if

$$n [ 1 - F (a_n x + b_n)] \Rightarrow w(x), \quad \text{then} \quad G_{m,n} (a_n x + b_n) \Rightarrow H_M (x).$$

Hence, by Corollary 6, H_m (x) is everywhere continuous, and thus

$$\lim_{n\to\infty} G_{m,n} (a_n x + b_n) = H_m(x)$$

for every $x \in E$. It follows from Theorem 11 that G_m,n (a_n x + b_n) converges uniformly to H_m (x) for $x \in E$.