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 $${\mathbb{E}}[(g_n-s_n)^2]={\cal{O}}(n^{-\alpha})$$

  • $\begingroup$ $n$ 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. $\endgroup$ May 26 '17 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..."? $\endgroup$ May 26 '17 at 2:31
  • $\begingroup$ Good point. Should read better now. $\endgroup$ May 26 '17 at 2:34
  • 1
    $\begingroup$ 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 $X$? $\endgroup$ May 26 '17 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. $\endgroup$ May 26 '17 at 20:22

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 $X$?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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