3
$\begingroup$

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

$\endgroup$
  • 1
    $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$

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.