Let $X_n$ be independent random variables which with probability $\frac 1 {2^{n+1}}$ is equal to $2^n$, with probability $\frac 1 {2^{n+1}}$ is equal to $-2^n$ and with probability $1 - \frac 1 {2^{n}}$ is equal to $0$.
Let $\overline{X}_k = \frac {\sum\limits_{n=1}^{k} X_n} {k}$
Are the conclusions of Law of large numbers and Central limit theorem true for $\overline{X}_k$?
By Borel-Cantelli, almost surely, only finitely many of $X_n$ are non-zero. Therefore, $$ a_n^{-1}\sum_{i=1}^n X_i $$ converges to zero almost surely (and thus in probability and in distribution) for any sequence $a_n$ that tends to infinity.