I consider Gabriel's example "Showing that Lindeberg condition does not hold" where we deal with a sequence of random variables $(X_n)_n$ with $X_k = Y_k + Z_k$ and $P(Y_k = \pm 1) = \frac{1}{2}$ and $$ P(Z_k = \pm k) = \frac{1}{2k^2} = \frac{1 - P(Z_k = 0)}{2}. $$ As in the solution there, it can be shown, that the CLT for $S_n = \sum_{k=1}^{n} X_n$ holds, that is $$ \frac{S_n}{\sqrt{n}} {\stackrel{d}{\longrightarrow}} \eta \sim \mathcal{N}(0,1) \quad \text{for } {n \rightarrow \infty} $$ but the Lindeberg condition is not fullfilled (even for the little more genereal Case where $Y_k$ is iid with expectation 0 and variance 1).
But I struggle with the Feller condition. As fare as I know this is (if I use the triangular array with $X_{n_k}:=(\frac{X_k}{\sqrt{n}})$): $$ \max_{1 \leqslant k \leqslant n} \sigma_{n_k} = \max_{1 \leqslant k \leqslant n} \mathbb{E}\left[\left(\frac{X_k}{\sqrt{n}}\right)^2\right] = \max_{1 \leqslant k \leqslant n} \frac{\mathbb{E}(X_k^2)}{n} = \frac{2}{n} \stackrel{n \to \infty}{\longrightarrow} 0, $$ as it may be found in mathworld. However, if the Feller condition is fulfilled, the Lindeberg condition is necessary and sufficient for the CLT.
Do I have a mistake with the Feller condition or is there another assumption for the equivalence Lindeberg $\Leftrightarrow$ Feller & CLT I do not considere/know?
And maybe another but minor important question: the example shows that the second moment of $\frac{S_n}{\sqrt{n}}$ does not converge. What do I need that $\mathbb{V}\left(\frac{S_n}{\sqrt{n}}\right) \stackrel{n \to \infty}{\longrightarrow} \mathbb{V}(\eta)$?
The Variance $\mathbb{V}\left(\frac{S_n}{\sqrt{n}}\right) = 2 =: \sigma_X^2$ is constant for every $n$. As Feller holds Lindeberg would imply that $\frac{S_n}{\sqrt{n}} \stackrel{d}{\rightarrow} Z \sim \mathcal{N}(0,\sigma_X^2)$ which is a contradiction to the limiting standard normal distribution. Therefore Lindeberg cannot be fullfilled.