Law of Large Numbers and Central Limit Theorem

Question

Decide whether the law of large numbers and central limit theorem holds for the mutually independent variable $X_k$ with the distribution defined as follows: $$P\left[{X_k=\pm2^k}\right]=2^{-(2k+1)}$$

Answer 1

Assume that $\mathbb P(X_k=x_k)=\mathbb P(X_k=-x_k)=u_k$, $\mathbb P(X_k=0)=1-2u_k$, for any nonnegative sequence $(x_k)$ and some sequence $(u_k)$ with values in $[0,\frac12)$ such that $\sum\limits_ku_k$ converges.

Then Borel-Cantelli lemma indicates that $X_k\ne0$ for only finitely many indexes $n$, almost surely. Setting $S_n=X_1+\cdots+X_n$, one sees that $(S_n)$ is almost surely bounded. In particular, $\frac1nS_n\to0$ almost surely (the law of large numbers holds) and $\frac1{\sqrt{n}}S_n\to0$ almost surely, hence in distribution (the central limit theorem holds, with a degenerate limit).

Note: (1) In the exercise $u_n=2^{-2n-1}$ hence $\sum\limits_ku_k$ converges. (2) The proof above does not use the independence property. (3) This assumes that $X_k$ is either $x_k$ or $-x_k$ or $0$, something which is not confirmed yet by the OP, despite several proddings to this effect in the comments.