Let $X_1,\ldots X_n$ be i.i.d. random variables and denote by $S_n$ their sum. Assuming that $\mathbb{E}S_n=0$, $\mathbb{E}S^2_n=1$ and that $\mathbb{E}|X_i|^3=b/n$, the Berry-Esseen Theorem (in the i.i.d. case) gives us the estimate

$$|P(Z<x)-P(S_n<x)|\leq cb,$$

where $c$ is an absolute constant and $Z$ has the standard normal distribution.

I wanted to know what happens if we do not assume finiteness of the $3-$rd or $(2+\varepsilon)$-th moments? Could it still be true that the latter inequality is still true provided we allow the constant to depend on the distribution of $X_1$? It is easy to construct examples of $2-3$ point distributions that would make that constant arbitrary large (for those examples). But once we fix the distribution, is it in general reasonable to assume that that the is a constant so that the aforementioned inequality holds?

Let $X_1,\ldots X_n$ be i.i.d. random variables and denote by $S_n$ their sum. Assuming that $\mathbb{E}S_n=0$, $\mathbb{E}S^2_n=1$ and that $\mathbb{E}|X_i|^3=b$, the Berry-Esseen Theorem (in the i.i.d. case) gives us the estimate

$$|P(Z<x)-P(S_n<x)|\leq cb/\sqrt{n},$$

where $c$ is an absolute constant and $Z$ has the standard normal distribution.

I wanted to know what happens if we do not assume finiteness of the $3-$rd or $(2+\varepsilon)$-th moments? Could it still be true that

$$|P(Z<x)-P(S_n<x)|\leq c'/\sqrt{n},$$

where $c'$ depends only on the distribution of $X_1$? 

It is easy to construct examples of $2-3$ point distributions that would make $c'$ arbitrary large (for those examples). But once we fix the distribution, is it in general reasonable to assume that that the is a constant so that the aforementioned inequality holds?