# CLT for non iid random variables

Assume $$U_k$$ are correlated standard normal random variables.

Let $$R_k := a_k U_k$$. I'm looking for CLT of the sum $$S_p := \sum_{k=1}^{p}\frac{R_k}{\sqrt{p}}$$.

Since $$U_k$$ are correlated, I'm looking at CLT for weakly correlated variables, but here identical distribution is also assumed, so not sure what to do with weights $$a_k >0$$ ($$\sum_{k=1}^{\infty}a_k < \infty$$). On the other hand, there are variants of CLT for non iid variables, but often independency is assumed.

Which CLT would work in my case? Are there any known results that would work under combined (weak) dependency and non-identical distribution?

## 1 Answer

The condition $$\sum a_k < \infty$$ seems too strong. It implies that $$S_p$$ converges in probability to 0, rather than converging in distribution to some nontrivial distribution.

• Thanks, I think we can relieve this condition. Are there known results I could try applying here? – runr Apr 3 at 12:21