Corollary 5 (modified by Gnedenko (1943), pp. 438-439)

Suppose that $\phi(x)$ is a proper frequency distribution. For any finite real constants a_n > 0 and b_n, $n = 1,2,3,\ldots$,

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

if and only if

$$\quad \lim_{n\to\infty} n [ 1 - F (a_nx + b_n)] = - \log \phi, \qquad \text{for all } x.$$

Proof:

(a)

If $F^n (a_n x + b_n) \Rightarrow 0(x)$, then by Corollary 4,

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

It follows from Theorem 5 that

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

(b)

If $\lim_{n\to\infty} n \left[ 1 - F (a_nx + b_n)\right] = -\log \phi (x)$ for all x, then by Theorem 5,

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

for all x. The limit holds then for the continuity points of $\phi(x)$, which are a subset of the points on the x axis, and

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