Theorem 9

Let H_m(x) and w(x) be, respectively, a proper frequency distribution and a proper W-function which possesses the interrelationship

$$H_m (x) = e^{-w(x)} \sum^{m-1}_{k=0} \frac{[w(x)]^k}{k!}$$

for m finite. Let a_n > 0 and b_n, $n = 1,2,3,\ldots$, be a sequence of real constants. Then,

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

if and only if for finite m,

$$G_{m,n} (a_n x + b_n) \Rightarrow H_m (x).$$

Proof: The theorem is an immediate consequence of Theorems 6 and 7.