Let $\{X_k\}$ are independent, $S_n = \sum_{k=1}^n X_k, D_n^2 = Var(S_n) < \infty$
(a) Show that if $\exists q >2$,s.t $$ \lim_{n \rightarrow \infty} D_n^{-q} \sum_{k=1}^n E\{ |X_k - EX_k|^q \} = 0 $$ then $D_n^{-1}(S)n - ES_n) \xrightarrow[]{d} N(0,1)$
(b)Show that part(a) applies in the case $D_n \rightarrow \infty$ and $E\{ |X_k - EX_k|^q \} \leq C \cdot Var (X_k)$ for some $q >2, C > 0, k = 1,2,\ldots$
I approach part (a) by showing that the Lyapunov condition implies the Lindeberg condition, but I have very little clue about part (b).
Any help would be greatly appreciated. Thank you.
In part (b), you have to check that $\lim_{n \rightarrow \infty} D_n^{-q} \sum_{k=1}^n E\{ |X_k - EX_k|^q \} = 0$. To do so, use the given bound to get $$ D_n^{-q} \sum_{k=1}^n E\{ |X_k - EX_k|^q \}\leqslant CD_n^{-q}\sum_{k=1}^n\operatorname{Var}(X_k)=CD_n^{2-q}. $$ Now you can conclude using the assumption on $D_n$ and $q$.