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Background: Let $\{a_i\}_{i=1}^n$ be i.i.d. random variables with zero-mean and unit variance, on a probability space $\Omega$. 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 an ${\cal{N}}(0,1)$ distributed random variable $g_n:\Omega\to\mathbb R$ so that $g_n,s_n$ 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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  • $\begingroup$ $n$ is fixed. I am not interested in a single $g$ that works for all $n$. $g$ only needs to be close to $s_n$ and $s_n$ only for that fixed $n$. Repeating the question: From distribution of $s_n$ (which is a fixed distribution), can you construct a joint distribution where the marginals give $g,s_n$ and $g$ is Gaussian, and $g-s_n$ has small second moment. $\endgroup$
    – ecstasyofgold
    Commented May 26, 2017 at 2:24
  • $\begingroup$ Ok, I think I see now. Would it be clearer to phrase it as "for each $n$ there exists $g_n$ such that..."? $\endgroup$
    – Nate Eldredge
    Commented May 26, 2017 at 2:31
  • $\begingroup$ Good point. Should read better now. $\endgroup$
    – ecstasyofgold
    Commented May 26, 2017 at 2:34
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    $\begingroup$ Did you try given $g_n$ to let $s_n=F_{s_n}^{-1}F_{g_n}(g_n)$ where $F_X$ is the c.d.f. of $X$? $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented May 26, 2017 at 6:53
  • $\begingroup$ Good point. Actually I figured out quite a bit of stuff after asking the question. Directly related to what you suggest, the problem is related to Monge–Kantorovich transportation: en.wikipedia.org/wiki/Transportation_theory_(mathematics) The following paper claims the best X,Y variables that minimize the distributional distance in Berry-Esseen is simply $F^{-1}_{s_n}(U)$ and $F^{-1}_{g_n}(U)$ where $U$ is uniform distribution which boils down to what you suggest I believe. Check out Eq (16) of arxiv.org/pdf/0906.5145.pdf. $\endgroup$
    – ecstasyofgold
    Commented May 26, 2017 at 20:22

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Did you try given $g_n$ to let $s_n=F_{s_n}^{-1}F_{g_n}(g_n)$ where $F_X$ is the c.d.f. of $X$?

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