Question about Feller's book on the Central Limit Theorem

My question concerns the proof of Theorem 1, section VIII.4, in Vol II of Feller’s book ‘An Introduction to Probability Theory and its Applications’. Theorem 1 proves the Central Limit Theorem in the i.i.d. zero mean, unit variance case.

In purely analytical terms, this theorem says that if $F$ is a (probability) distribution function with zero expectation and unit variance, then $$F^{n*}(x\sqrt{n}) \rightarrow \mathfrak{N}(x)$$ in the distributional sense (where $\mathfrak{N}$ is the standard normal distribution, and $(\cdot)^{n*}$ denotes the n-fold convolution operation).

To prove this, Feller uses the following (the proof of which I understand):

$\textbf{Lemma}$ $\quad$ Denote by $\mathfrak{F}_n$ the convolution operator associated with the functions $F_n(x):= F(x\sqrt{n})$. Then for any $u : \mathbb{R} \rightarrow \mathbb{R}$ with three bounded derivatives and finite limits at $\pm \infty$, we have the following uniform convergence on the line: $$n[\mathfrak{F}_nu – u] \rightarrow \frac{1}{2}u”. \quad \square $$

Now denote by $\mathfrak{G}$ and $\mathfrak{G}_n$ resoectively the convolution operators associated with the distributions $\mathfrak{N}(x)$ and $\mathfrak{N}(x\sqrt{n})$. Then it is easy to derive the following inequality, where the norm is the supremum norm:

$$ \|\mathfrak{F}_n^n(u) – \mathfrak{G}(u) \| \leq n\|\mathfrak{F}_n(u) – u \| + n\|\mathfrak{G}_n(u)-u \| .$$

Now comes the bit I don’t understand. He claims that, by the Lemma above, the RHS of this inequality tends to zero. (N.B. He doesn’t specify what kind of function $u$ is). How can this be so, unless $u”=0$ on the line? This suggests to me that he is approximating all continuous functions on $\mathbb{R}$ with finite limits at $\pm \infty$ by functions with zero second derivative. I guess this can be done? And do you think this is what Feller is doing at this stage?

Many thanks for your help.

Frank.

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I know i’m probably 3 years late, but i was studying this chapter of the book and hammering my head around this lemma and theorem for half a day before I managed to make sense of it!

In my opinion, i think there is a small mistake on his proof. I would go on to prove it like this:

$$||\frak{F}_n^n u – \frak{G}u|| = ||\frak{F}_n^n u – \frak{G}_n^n u|| \leq n||\frak{F}_n u – \frak{G}_n u|| = ||n(\frak{F}_n u – u) – n(\frak{G}_n u – u)||\to\left|\left|\frac{u^{\prime \prime}}{2} – \frac{u^{\prime \prime}}{2}\right|\right|$$

Then the lemma works like a glove to finish the proof