Let $\; X_0,X_1,X_2,X_3,...\;$ be independent and identically distributed (real-valued) random variables.

1. Suppose $\frac1n \cdot\sum\limits_{m=0}^n X_m$ converges in probability. Does it follow that $\operatorname{E}(X_0)$ exists?

2. Suppose $\operatorname{E}(X_0) = 0$ and that $\frac1{\sqrt n} \cdot\sum\limits_{m=0}^n X_m$ converges in distribution to a normal random variable. Does it follow that $\operatorname{E}((X_0)^2)$ is finite?

(I already found that the converse of the strong law of large numbers holds.)