Corollary 2 (Gnedenko (1943), p. 438)

  If there exist finite real constants a_n > 0 and b_n such that

 

$$F^n (a_n x + b_n) \Rightarrow \phi(x), \qquad n \to \infty,$$ (10)

where $\phi(x)$ is a proper frequency distribution, then

$$\lim_{n\to\infty} \frac{a_n}{a_{n+1}} = 1 \quad \text{and} \quad \lim_{n\to\infty} \frac{b_n - b_{n+1}}{a_n} = 0.$$

Proof: Let x₀ be any continuity point of $\phi(x)$. Then,

 

$$F^{n+1} (a_{n+1} x_0 + b_{n+1}) \Rightarrow \phi(x_0), \quad \text{as } n \to \infty.$$ (11)

If $\phi(x_0) \neq 0$, then $\lim_{n\to\infty} F(a_n x_0 + b_n) = 1$.If $\phi (x_0) = 0$, then $\lim_{n\to\infty}$ $F(a_n x_0 + b_n) \leq 1$,while $\lim_{n\to\infty} F^n (a_n x_0 + b_n) = 0$. In either case,

 

$$\lim_{n\to\infty} F(a_n x_0 + b_n) F^n (a_n x_0 + b_n) = \lim_{n\to\infty} F^{n+1} (a_n x_0 + b_n) = \phi (x_0).$$ (12)

Equations (11) and (12), together with Theorem 2, prove the corollary.