Theorem 6

Let $\phi(x)$ and H_m (x) be real functions such that

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

Then, for any finite real constants a_n > 0 and b_n, $n = 1,2,3,\ldots$,

$$F^n (a_n x + b_n ) \Rightarrow \phi (x) \qquad (\phi(x) \text{ a proper frequency distribution}),$$

if and only if

$$G_{m,n} (a_n x + b_n) \Rightarrow H_m (x) \qquad (H_m(x) \text{ a proper frequency distribution}).$$

Proof:

(a) If $F^n (a_n x + b_n) \Rightarrow 0(x)$, then by Theorem 4 for all x,

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

Also, Corollary 5 gives that

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

Hence, by Theorem 17, assuming that $\lim_{n\to\infty} F (a_n x + b_n) = 1$,

 

$$\begin{array} {r@{~}c@{~}l} \displaystyle \lim_{n\to\infty} ... ...m^{m-1}_{k=0} \frac{[-\log 0(x)]^k}{k!} = H_m (x). \end{array}$$ (35)

If $\overline{\lim} _{n\to\infty} (F(a_n x + b_n) < 1$, choose $\epsilon$ so that $\overline{\lim} _{n\to\infty} F(a_n x + b_n) < \epsilon < 1$. Then, for n large enough,

$$\begin{array} {r@{~}c@{~}l} G_{m,n} (a_n x+b_n) & = & \displ... ...n{pmatrix} n \\ k \end{pmatrix} \epsilon^{n-k} n^k.\end{array}$$

Hence (Rudin (1953), p. 43, theorem 3.20(d)),

 

$$\begin{array}[t] {c} \overline{\lim} \\ \scriptstyle n\to\i... ...ft( 1 - \frac{k-1}n\right) }{k! \epsilon^k} \epsilon^n n^k = 0.$$ (36)

Now, by Theorem 18, together with the relationship between H_m(x) and $\phi(x)$ given in the hypothesis,

 

$$\begin{array} {r@{~}c@{~}l} H_m (x) & = & \displaystyle s^{\... ...{\log \phi(x)}_{0} \frac{e^{-s}s^{m-1} ds}{(m-1)!}. \end{array}$$ (37)

Since I_x(m) is the gamma frequency distribution [Cramér (1946), p. 126, eq. (12.3.3)], it follows that H_m (x) = 0 when $\phi(x) = 0$. Furthermore, if $\overline{\lim} _{n\to\infty} F (a_n x+b_n) < 1$, then

 

$$\lim_{n\to\infty} F^n (a_n x + b_n) = \phi (x) = 0.$$ (38)

Hence, combining Eqs. (36) and (38), if $\overline{\lim} _{n\to\infty} F (a_n x+b_n) < 1$, then $\phi(x) = 0$, H_m (x) = 0 and

 

$$\begin{array}[t] {c} \overline{\lim} \\ \scriptstyle n \to \infty \end{array}G_{m,n} (a_n x + b_n) = 0 = H_m (x).$$ (39)

Since I_x(m) is a frequency distribution, Eq. (37) shows that H_m (x) is also a frequency distribution. If $\phi(x)$is proper, then there exists x₀ such that $0 < \phi (x_0) < 1$.Hence, by Eq. (37), 0 < H_m (x₀ ) < 1 and H_m (x) is proper.

Thus, in all cases for all x,

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

and certainly the weaker relationship

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

holds.

(b) Now, suppose that

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

where H_m (x) is a proper frequency distribution. Let x₀ be any continuity points of H_m (x). Since $0 \leq F^n (a_n x_0 + b_n) \leq 1$ for all values of n, it follows from the Bolzano-Weierstrass theorem [Rudin (1953), pp. 31-32, theorem 2.37] that the sequence $\left\{ F^n (a_n x_0 + b_n ) \right\}$ has one or more limit points. Assume that $\left\{ F^n (a_n x_0 + b_n ) \right\}$ has at least two limit points, and let $0 \leq A < B \leq 1$be any two such limit points. Let $r_1 < r_2 < r_3 < \ldots$ and $s_1 < s_2 < s_3 < \ldots$ be two subsequences of the positive integers such that

$$\begin{array} {r@{~}c@{~}l} \displaystyle \lim_{k\to\infty} ... ...\to\infty} F^{s_k} (a_{s_k} x_0 + b_{s_k}) & = & B. \end{array}$$

Now, by reasoning identical with that used in obtaining Eqs. (35) and (39),

$$\begin{array} {r@{~}c@{~}l} \displaystyle \lim_{k\to\infty} ... ...playstyle B \sum^{m-1}_{k=0} \frac{[-\log B]^k}{k!}.\end{array}$$

Then, by Theorem 18 and Definition 9,

 

$$\begin{array} {r@{~}c@{~}l} \displaystyle \lim_{k\to\infty} ... ...- \int^{-\log B}_0 \frac{e^{-s} s^{m-1}ds}{(m-1)!}. \end{array}$$ (40)

Now,

$$\frac{d}{dx} \int^x_0 \frac{e^{-s}s^{m-1}ds}{(m-1)!} = \frac{e^{-x} x^{m-1}}{(m-1)!} \gt 0, \qquad \text{for } 0 \leq x,$$

and so $1 - \int^x_0 (e^{-s} s^{m-1} ds)/(m-1)!$ is a strictly monotonic decreasing function of x. It follows that if, as assumed,

$$0 \leq A < B \leq 1,$$

then

$$0 \leq \log B < \log A \leq \infty$$

and so

$$\lim_{k\to\infty} G_{m,r_k} (a_{r_k} x_0 + b_{r_k}) < \lim_{k\to\infty} G_{m,s_k} (a_{s_k} x_0 + b_{s_k}).$$

This contradicts the assumption that

$$\lim_{n\to\infty} G_{m,n} (a_n x_0 + b_n) = H_m (x_0).$$

Hence, the sequence $\left\{ F^n (a_n x_0 + b_n ) \right\}$ has a unique limit point for every continuity point x₀ of H_m(x), and by Eq. (40) if $\phi (x_0)$ denotes this unique limit point,

$$H_m (x_0) = 1 - \int^{-\log \phi (x_0)}_0 \frac{e^{-s}s^{m-1}ds}{(m-1)!}.$$

Define $\phi(x)$ by the equation

 

$$H_m (x) = 1 - \int^{-\log \phi (x)}_0 \frac{e^{-s} s^{m-1}ds}{(m-1)!}.$$ (41)

It has already been demonstrated that for every continuity point x₀ of H_m (x),

 

$$\lim_{n\to\infty} F^n (a_n x_0 + b_n) = \phi (x_0).$$ (42)

It remains to show that

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

for every continuity point of $\phi(x)$ and that $\phi(x)$ is a frequency distribution. The last relation will be considered first. Since H_m(x) is a strictly monotonic increasing function of $\phi(x)$, the converse is also true, and $\phi(x)$ is a strictly monotonic increasing function of H_m(x). But, H_m(x) is a monotonic increasing function of x, and so $\phi(x)$ is a monotonic increasing function of x. By inspection of Eq. (41), $\phi(x)$ and H_n(x) are zero together and one together. Hence,

$$\begin{array} {r@{~}c@{~}l} \displaystyle \lim_{x\to\infty} ... ...\\ \displaystyle \lim_{x\to\infty} \phi (x) & = & 0.\end{array}$$

Thus, $\phi(x)$ is a frequency distribution. Also, if 0 < H_m(x) < 1, then $0 < \phi (x) < 1$. It follows that if H_m(x) is proper, then $\phi(x)$ is proper.

Now, the continuity relationship will be considered. It will be demonstrated that every continuity point of $\phi(x)$ is a continuity point of H_m(x). Suppose that $\phi(x)$ is continuous at x = x₁. Choose any $\epsilon \gt 0$. Now, if $\phi(x)$ is continuous at x = x₁, then $\log \phi (x)$ is continuous at the same point. Hence, there exist $\delta \gt 0$ such that if $\vert x - x_1\vert < \delta$,then $\vert \log \phi(x) - \log \phi (x_1)\vert < \epsilon$. Now, since

$$e^s = 1 + s + \frac{s^2}{2!} + \ldots + \frac{s^{m-1}}{(m-1)... ... \ldots \gt \frac{s^{m-1}}{(m-1)!}, \qquad \text{for } s \gt 0,$$

it follows that

$$\frac{e^{-s} s^{m-1}}{(m-1)!} \leq 1, \qquad \text{for } 0 \leq s < \infty.$$

Hence,

$$\begin{array} {r@{~}c@{~}l} \left\vert H_m (x) - H_m (x_1) \... ...g \phi (x_1) - \log \phi (x) \right\vert < \epsilon,\end{array}$$

and H_m(x) is continuous at x = x₁. But then, by Eq. (40),

$$\lim_{n\to\infty} F^n (a_n x_1 + b_n) = \phi (x_1),$$

and it follows that

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

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