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The central limit theorem states that, if $X$ is a random variable with finite variance $\sigma^2$ and expected value $\mu$, and if $(X_n)$ is a sequence of independent random variables identically distributed like $ X $, then

\begin{equation} Z_n = {\frac{{\overline{X}}_n-\mu}{\sqrt{\sigma^2/n}}}\ \rightsquigarrow N(0,1), \end{equation}

where ${\overline{X}}_{n} = \frac{1}{n} \sum_{i=1}^{n} X_i$ and $\rightsquigarrow$ means convergence in distribution, which in this case is equivalent to the pointwise convergence of the cdf of $Z_n$ to the cdf of a $N(0,1)$.

Suppose moreover that $X_n$ is absolutely continuous for all $n$‘s. Then $Z_n$ is absolutely continuous for all $n$‘s. Are there examples of such sequences $(X_n)$, such that the pdf of $Z_n$ does not converge pointwise to the pdf of a $N(0,1)$, 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. $\endgroup$
    – user140541
    Aug 23, 2019 at 23:33
  • $\begingroup$ @d.k.o. Could you elaborate? $\endgroup$
    – Federico
    Aug 23, 2019 at 23:54
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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). $\endgroup$
    – user140541
    Aug 24, 2019 at 2:00
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    $\begingroup$ @MathieuKrisztian, no, they don’t. They just provide a counterexample that shows that the independence hypothesis in the theorem cannot be replaced with the weaker pairwise independence. $\endgroup$
    – Federico
    May 2, 2021 at 20:04
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    $\begingroup$ @MathieuKrisztian, the CLT’s hypotheses are not satisfied by any sequence of independent random variables. In the version of the CLT that I mentioned (the classical one) it is required that the random variables are also identically distributed. There are other versions of the theorem that require the sequence to satisfy different conditions, but you can’t expect to take any sequence of independent random variables and have their normalized mean converge (weakly) to a $N(0,1)$. $\endgroup$
    – Federico
    May 2, 2021 at 20:41

3 Answers 3

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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.

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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. $\endgroup$ Aug 24, 2019 at 10:58
  • $\begingroup$ jstor.org/stable/pdf/… $\endgroup$
    – Myshkin
    Sep 11, 2019 at 21:38
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Let $X\sim\mathcal N(\mu,\sigma^2)$ and $Y=1/X$. Then $$ \frac{\frac{1}{n}\sum_{k=1}^nY_k-\mathsf E_\mathcal PY}{\pi f_X(0)}\overset{d}{\to}\operatorname{Cauchy(0,1)}, $$ where $\mathsf E_\mathcal PY$ denotes the cauchy principal value integral $$ \mathsf E_\mathcal PY:=\mathcal P\int_{-\infty}^\infty y f_Y(y)\,\mathrm dy, $$ and $\operatorname{Cauchy(0,1)}$ denotes the standard Cauchy distribution. For this reason, we say $Y$ lies within the domain of attraction of the Cauchy law as opposed to the normal (Gaussian) law which is used in the CLT.

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I can recommend

A Few Counter Examples Useful in Teaching Central Limit Theorems., www.jstor.org/stable/24590348.

As well as chapter 17 of the great book Counterexamples in Probability by Jordan Stoyanov.

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