In Varadhan's probability book it is stated that there exists sequences of identically distributed uncorrelated (finite variance?) RV's such that the central limit theorem does not hold. The book doesn't cite an example though. I was wondering if such an example is easy to construct.
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3 Consider the random variables $X_n=Y_n\cdot Z$ where the sequence of random variables $(Y_n)_n$ is i.i.d. square integrable and centered, and where the random variable $Z$ is square integrable and independent of $(Y_n)_n$. Then the sequence of random variables $(X_n)_n$ is square integrable, centered and uncorrelated.
The central limit theorem applied to the sequence $(Y_n)_n$ shows that the random variables $\frac1{\sqrt{n}}(X_1+\cdots +X_n)$ converge in distribution to $YZ$, where the random variable $Y$ is standard normal and independent of $Z$.
When the random variable $Z$ is not almost surely constant, there is no reason to expect the random variable $YZ$ to be normal. For example, if $Z$ is a Bernoulli random variable with $P(Z = 0) = P(Z = 1) = \frac{1}{2}$, then $YZ$ is non-normal, specifically a mixture of a normal distribution and a point mass at zero. Thus, convergence in distribution holds, but not necessarily to a normal distribution.
- $\begingroup$ This would be more complete if an instance were exhibited of distributions of $Y_n$ and of $Z$ for which one would prove that $YZ$ is not normally distributed. ${}\qquad{}$ $\endgroup$Michael Hardy– Michael Hardy2015-09-21 16:17:27 +00:00Commented Sep 21, 2015 at 16:17
- 4$\begingroup$ @MichaelHardy This is quite complete as it stands, thank you. If you have trouble finding an example, be reassured that nearly everything works (say, Z asymmetric Bernoulli). O wait! This was already clear from the answer, right? $\endgroup$Did– Did2015-09-21 17:28:20 +00:00Commented Sep 21, 2015 at 17:28
- 1$\begingroup$ If only math.stackexchange would have a like button, I would have liked the latter comment. Things being what they are, I could only rank it up. $\endgroup$TOM– TOM2015-11-22 15:44:45 +00:00Commented Nov 22, 2015 at 15:44