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