# Central limit theorem and integrability

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?

• 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. – Jason Swanson Apr 25 at 21:52