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?



Asymptotic rate for the expected value of the square root of sample average
Substituting $S\_n$ for $u$ in the inequalities 
$$\frac{1+u-(u-1)^2}2\le\sqrt u\le\frac{1+u}2$$
for $u\ge0$, taking the expectations, and using that $ES\_n=1$ and $E(S\_n-1)^2=V(S\_n)=\sigma^2/n$, we have 
$$1-\frac{\sigma^2}{2n}\le E\sqrt{S\_n}\le1,$$
so that $E\sqrt{S\_n}=1-O(1/n)$, as desired. 


