Corollary 4

  If there exist real finite constants a_n > 0 and b_n, $n = 1,2,3,\ldots$, such that

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

where $\phi(x)$ is a proper frequency distribution, then $\phi(x)$is continuous for all x. Hence, for all x,

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

and the concept of weak convergence may be replaced with the usual type of limiting process [see Rudin (1953), for instance].

Proof: The corollary follows immediately from the properties of the limiting types permitted by Theorem 4.