Background: Let $\{a_i\}_{i=1}^n$ be i.i.d. random variables with zero-mean and unit variance. Define $$s_n=\frac{1}{\sqrt{n}}\sum_{i\leq n} a_i$$

Central limit theorem says that the distribution of $s_n$ converges to the standard normal ${\cal{N}}(0,1)$.

Short question: For each $n$, give me a Gaussian random variable $g_n$ that is close to $s_n$.

Rigorous question: Fix the number $n$. Can we construct a Gaussian ${\cal{N}}(0,1)$ distribution from $s_n$'s distribution so that if $(s_n,g_n)$ is sampled from the joint distribution we constructed (where $g_n$ is the Gaussian variable) $g_n,s_n$ are close in some sense. For instance, for some $\alpha>0$, we want $${\mathbb{E}}[(g_n-s_n)^2]={\cal{O}}(n^{-\alpha})$$

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    $\begingroup$ what do you mean "construct a gaussian from $s_n$'s distribution".. $\endgroup$ – user408433 May 26 '17 at 4:09
  • $\begingroup$ This is very unclear. You want $g$ such that the normed sums converge to it? This is impossible. Maybe a sequence $g_n$ such that $s_n - g_n\to0$? $\endgroup$ – zhoraster May 26 '17 at 6:04

You are basically asking about a central limit theorem with respect to the Wasserstein 2 metric (you can read more about it here https://en.m.wikipedia.org/wiki/Wasserstein_metric).

Formally, it is equivalent to the classical CLT but it requires more finness to actually find the coupling. You can see some recent results here https://arxiv.org/abs/1506.06966.


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