Give an example where there exist $C>0, q>2$ such that $\mathbf{E}|X-\mathbb{E}X_k|^{q}\leq C\text{Var}(X_k)^{q/2}$ for all $k$ and $\sigma_n\rightarrow\infty$, yet $(S_n-\mathbb{E}S_n)/\sigma_n$ doesn't converge in distribution.
Note: $S_n=X_1+...+X_n$ where $X_i$ are r.v, $\sigma_n^2=VarS_n<\infty$
I have been thinking about this for a while anyone have a good and easy example?
Define $U$ as a random variable that is uniform over $[-1,1]$. Define $Z$ as an independent Gaussian with zero mean and unit variance. Define: $$ X_n = \left\{\begin{array}{ll} 2^n U & \mbox{if $n$ is even} \\ 2^n Z & \mbox{if $n$ is odd} \end{array} \right. $$ The idea is that $S_n/\sigma_n$ is dominated by random variable $X_n$, whose distribution keeps changing. To make it more obvious, you could define $X_n$ as $2^{2^n}U$ if $n$ is even and $2^{2^n}Z$ if $n$ is odd. You can also make the $\{X_n\}_{n=1}^{\infty}$ independent if you want, using a similar trick.
