Let X1, X2, . . . be independent identically distributed random variables with P(X1 = 1) = P(X1 = −1) = 1/2. For any n ≥ 1, define
$S_n = \frac{X_1+...+X_2}{\sqrt{n}}$
The Central Limit Theorem says that Sn converges in distribution to a standard Gaussian random variable. We show that Sn does not converge in probability to any random variable. The intuition here is that if Sn did converge in probability to a random variable Z, thenwhen n is large, Sn is close to Z, $Y_n = \frac{\sqrt{2}S_{2n} - S_n}{\sqrt{2}-1}$ is close to Z, but Sn and Yn are independent. And this cannot happen. Proceed as follows. Assume that Sn converges in probability to Z.
• Let ε > 0. For n very large (depending on ε), we have P(|Sn − Z| > ε) < ε and P(|Yn − Z| > ε) < ε.
• Show that P(Sn > 0, Yn > 0) is around 1/4, using independence and the Central Limit Theorem.
• From the first item, show P(Sn > 0|Z > ε) > 1 − ε, P(Yn > 0|Z > ε) > 1 − ε, so P(Sn > 0, Yn > 0|Z > ε) > 1 − 2ε.
• Without loss of generality, for ε small, we have P(Z > ε) > 4/9.
• By conditioning on Z > ε, show that P(Sn > 0, Yn > 0) is at least 3/8, when n is large.
I understand in the first part that you multiple the 2 probabilities because they are independent. Don't know how to get the results I want though. I'm stuck with the rest too.