Precision of Sum of Random Variables

If we increase the sample size on the average by a factor k, then the precision 1/(standard error) on the estimate of the mean increases by a factor of sqrt(n). My question is what if anything can we say quantitatively about the change in precision on the SUM of random variables as the sample size is increased? The variance of the sum increases by the factor k. Are there any convergence theorems that deal with this issue?

Thank you very much,



TS Contributor
Whenever you are dealing with the sum of random variables, if it satisfy the regularities of the Central Limit Theorem, then yes you can say that the order of convergence is \( \mathcal{O}_p\left(\frac {1} {\sqrt{n}}\right) \)