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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):

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$\textbf{Lemma}$ $\quad$ 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

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