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.)



Do the converses of [weak law of large numbers / central limit theorem] hold?
(As suggested, I promote my comment to an answer, with pgassiat's complement.)


Necessary and sufficient conditions (in terms close to those you want) for the WLLN and the CLT can be found, e.g., in "Foundations of modern probability" by Kallenberg (Theorems 4.16 and 4.17 in the first edition, Theorems 5.16 and 5.17 in the second edition). 


