Let $\{\{X_{n1},\ldots,X_{nn}\},n=1,2,\ldots\}$ be a triangular array of independent random variables where each row contains identically distributed random variables. Let $E[X_{n1}]=\mu_n<\infty$ and $E[(X_{n1}-\mu_n)^2]=\sigma_n^2<\infty$. Furthermore, let $E[|X_{n1}-\mu_n|^3]=\rho_n<\infty$.

Let $S_n=\sum_{i=1}^nX_{ni}$ be the sum of row $n$. It's well-known that the central limit theorem holds in this case: $\frac{S_n-n\mu_n}{\sqrt{n}\sigma_n}\xrightarrow{\mathcal{D}} Z$ where $Z\sim\mathcal{N}(0,1)$ -- the proof can be found in, say, Billingsley Ch. 27.

I am wondering what is known about the rate of convergence in this scenario. I am hoping for a Berry-Esseen type of result, where $|F_n(x)-\Phi(x)|\leq \frac{C\rho_n}{\sigma^3_n\sqrt{n}}$ with $F_n(x)=P\left(\frac{S_n-n\mu_n}{\sqrt{n}\sigma_n}\leq x\right)$, however, I cannot find it. Does anything of this nature exist? I would be happy with a convergence rate that is $\mathcal{O}\left(\frac{1}{n^a}\right)$ with $0<a\leq\frac{1}{2}$, and am willing to impose constraints on higher-order absolute moments.

I did notice the discussion on this question, however, my scenario is simpler, and might be tractable (since the random variables are independent). Can anyone help?

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    $\begingroup$ Usually the Berry-Esseen is stated in a way which makes it clear that it applies to triangular arrays that have IID rows. See e.g. Durrett's "Probability: Theory and Examples", Theorem 4.9. $\endgroup$ – Bill Johnson Jan 15 '14 at 21:36
  • $\begingroup$ That's a nice, readable proof of the Berry-Esseen theorem. Thanks! I wasn't aware of that book, thanks for the pointer. $\endgroup$ – Bullmoose Jan 15 '14 at 23:29

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