The central limit theorem asserts that the normalized sum of a sequence of i.i.d. random variables X_1, X_2,..., with finite variance converges in distribution to a normal distribution. Moreover, there is a result by Ranga Rao which guarantees that if X_i has a pdf, then the sequence of pdf's converges to the pdf of a normal distribution almost everywhere.
I would like to know if a similar result is known for Lindeberg's Central Limit Theorem (where the random variables are independent but not identically distributed). In other words, I would like to know whether, if you have a sequence of independent random variables X_1, X_2, ..., which satisfies Lindeberg's condition and each of them has a pdf, you can guarantee that the pdf of the normalized sum converges almost everywhere to the pdf of a normal distribution.
I've been looking for the answer to this in many books, but I can't find it. Any suggestion will be appreciated!
Thanks!
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Actually, I found a counter-example, so the pdf of the normalized sum does not need to converge almost everywhere to the pdf of a normal distribution.