If $(Y_n)_n$ is a sequence of independent random variables and identically distributed, and if $\frac{\sum_{k=1}^nY_k}{\sqrt{n}}$ converges in distribution to a random variable Y, does this mean that $Y_1 \in L^2?$

I don't have any idea how to begin, and the result may appear as a converse for the central limit theorem.

Do you have any idea or a reference for this theorem?

  • $\begingroup$ This is Exercise 3.4.3 in Probability: Theory and Examples (4th edition) by Durrett. The text has a hint/sketch for you to follow, and there is also a solutions manual that you can reference if needed. $\endgroup$ – Jason Swanson Apr 25 at 21:52

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