The central limit theorem states that, if $ X $ is a random variable with finite variance and $ \{ X_{n} \}_{n=1}^{+\infty} $ is a sequence of independent random variables identically distributed like $ X $, then

\begin{equation} Z_{n} = {\frac{{\overline{X}}_{n}-E(X)}{\sqrt{var(X)/n}}}\ \xrightarrow{d}\ N(0,1), \end{equation}

where $ {\overline{X}}_{n} = {\frac{1}{n}} \sum_{j=1}^{n} X_{j} $ and $ \xrightarrow{d} $ means convergence in distribution. In this case, this means that the cdf of $ Z_{n} $ converges pointwise everywhere to the cdf of a $ N(0,1) $ as $ n \to +\infty $.

Suppose moreover that $ X_{n} $ is absolutely continuous for all $ n \geq 1 $. Then $ Z_{n} $ is absolutely continuous for all $ n \geq 1 $.

My question is: what are examples of sequences $ \{ X_{n} \}_{n=1}^{+\infty} $, described as above, such that the pdf of $ Z_{n} $ doesn't converge pointwise to the pdf of a $ N(0,1) $ as $ n \to +\infty $, not even almost everywhere (w.r.t. the Lebesgue measure on $ \mathbb{R} $)?

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    $\begingroup$ Such $\{X_n\}$ should violate the assumptions of the CLT. $\endgroup$ – d.k.o. Aug 23 at 23:33
  • $\begingroup$ @d.k.o. Could you elaborate? $\endgroup$ – Federico Aug 23 at 23:54
  • $\begingroup$ If $\{X_n\}$ is required to be a sequence of i.i.d. r.v.s with finite second moment, then by the CLT the self-normalized sum converges weakly to $N(0,1)$. $\endgroup$ – d.k.o. Aug 24 at 0:05
  • $\begingroup$ @d.k.o. I agree, but weak convergence means pointwise convergence of the cdfs to the cdf of the limit at all points of continuity for the cdf of the limit, and doesn't say anything about convergence of the pdfs in general. I don't think this case is any different, but please correct me if I'm wrong. $\endgroup$ – Federico Aug 24 at 0:11
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    $\begingroup$ I see. I missed that you're asking about pdfs. Indeed, weak convergence does not always imply convergence of densities (though, the opposite is true). $\endgroup$ – d.k.o. Aug 24 at 2:00

This counterexample is from the book Limit Distributions for Sums of Independent Random Variables by Gnedenko and Kolmogorov.

Let $X$ have density $\begin{cases} 0 &\text{if} |x|\geq \frac 1e \\ \frac{1}{2|x|\log^2(|x|)} &\text{if} |x|< \frac 1e \end{cases}$

The authors argue that $f_n$ the density of $\sum_{i=1}^n X_i$ verifies $\displaystyle f_n(x) > \frac{c_n}{|x \log^{n+1}(|x|)|}$ for some positive constant $c_n$ in a neighborhood of $0$. So the density of $Z_n$ (which is a normalized version of $f_n$) is infinite at $0$.

They prove the following theorem:

Theorem: Suppose $X$ has density $f$. If

  • for some $m\geq 1$, $f_m$ (the density of $\sum_{i=1}^m X_i$) is in $L^r(\mathbb R)$ for some $r\in (1,2]$,
  • $\int x^2 f(x) dx <\infty$ (i.e. $X$ has a second moment)

Then $\displaystyle \sup_{x\in \mathbb R} \left|\sigma \sqrt n f_n(\sigma \sqrt n x) - \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \right| \xrightarrow[n\to \infty]{}0$

In Petrov's Sums of Independent Random Variables, the following theorem is stated:

Theorem: Let $(X_n)$ be a sequence of i.i.d r.v with mean zero and variance $\sigma^2$ and let $f_n$ denote the density of $Z_n$ (if it exists).

Then $\displaystyle \sup_{x\in \mathbb R} \left| f_n(x) - \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \right| \xrightarrow[n\to \infty]{}0$ if and only if $f_n$ is bounded for some $n$.

In Shiryaev's Probability 2, the following Local Central Limit Theorem is stated:

Theorem: Let $(X_n)$ be a sequence of i.i.d r.v with mean zero and variance $\sigma^2$. If for some $r\geq 1$, $\int |\phi_{X_1}(t)|^r dt <\infty$, then $Z_n$ has a density $f_n$ such that $\displaystyle \sup_{x\in \mathbb R} \left| f_n(x) - \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \right| \xrightarrow[n\to \infty]{}0$

Regarding almost sure convergence, you should have a look at Rao's A Limit Theorem for Densities.

  • $\begingroup$ Thank you so much for including references! I'm having trouble getting my hands on Rao's paper since it doesn't seem I can get it via my university, hope I will make it eventually. $\endgroup$ – Federico Aug 24 at 10:50
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    $\begingroup$ @Federico I haven't been able to access it either, and unfortunately it seems Jstor is the only place where it's available... Petrov says a few words about Rao's paper, see page 213. $\endgroup$ – Gabriel Romon Aug 24 at 10:58
  • $\begingroup$ jstor.org/stable/pdf/… $\endgroup$ – miosaki Sep 11 at 21:38

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