Let $(X_n)_n$ be a sequence of $q$-dependent random variables and identically distributed. If $E[X_1^2]<+\infty,$ then the Hoeffding-Robbins theorem states that $$\frac{1}{\sqrt{n}}\sum_{k=1}^n(X_k-E[X_k]) \Rightarrow N(0,\sigma^2).$$

The converse seems to be true, that is if $\frac{1}{\sqrt{n}}\sum_{k=1}^nX_k$ converges in distribution to a random variable $X$ then $E[X_1^2]<+\infty,$ in particular when $q=0$: https://math.stackexchange.com/questions/3202451/central-limit-theorem-and-integrability/3623412#3623412 and https://math.stackexchange.com/questions/3961865/central-limit-theorem-for-weighted-random-variable

The converse, does it hold if $q>0$ ? Is there any proof, paper, reference... for this?


1 Answer 1


Probably not, there are trivial case like $X_1, -X_1, X_2, -X_2,...$ with the X's symmetric to make them identically distributed, but even if you have some good way to exclude them there are probably subtler cases where only the q-fold sums are at all well behaved.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.