I have iid random variables $X_1, \dots, X_n$ with $X_i \geq 0$, $E[X_i]=1$ and $V[X_i] = \sigma^2$. Let $S_n = \frac{\sum_{i=1}^n X_i}{n}$.

I'd like to say that $E[\sqrt{S_n}] = 1-O(1/n)$.

My first approach was to write $E[\sqrt{S_n}] = \sqrt{E[S_n] - V[\sqrt{S_n}]} = \sqrt{1-V[\sqrt{S_n}]}$.

I'm then left with showing that $V[\sqrt{S_n}] = O(1/n)$.

I'm unsure how to go about this. First, can I hope to prove such an asymptotic bound in general? If not, are there extra assumptions that can be made on the $X_i$ so that this holds true?

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