Math 6040 • Probability

Let \(\{X_{n}\}_{n \in \mathbb{N}}\) be independent random variables such that for each \(n\), \(X_{n}\) takes the
values \(+1\) and \(-1\) with probability \((1 - 2^{-n})/2\) each and the value \(2^{k}\) with probability
\(2^{-k}\), for integers \(k > n\).

\begin{enumerate}

\item
Show that \(\mathbb{E}[|X_{n}|] = \infty\) for all \(n \in \mathbb{N}\).



\item
Show that \((X_{1} + \cdots + X_{n})/\sqrt{n}\) converges in distribution to a standard normal.
(Hint: Check that we can write \(X_{n} = (1 - B_{n})Y_{n} + 2^{n}B_{n}Z_{n}\), where \(B_{n}\) is
\(\operatorname{Bernoulli}(2^{-n})\), \(Y_{n}\) takes values \(\pm 1\) equally likely, \(Z_{n}\) takes the value \(2^{k}\) with
probability \(2^{-k}\), for \(k \in \mathbb{N}\), and \(\{B_{n}, Z_{n}, Y_{n} : n \ge 1\}\) are mutually independent.
Now, \(X_{n}\) is a small perturbation of \((1 - B_{n})Y_{n}\). Work out the details.)

\end{enumerate}