Central limit theorem and the law of large numbers in the mean

Let $\{\xi_{n1},\xi_{n2},…,\xi_{nk_n}\}_{n=1}^{\infty}$ be a sequence of independent (for every $n\ge 1$) infinitesimal random variables. We prove that
$$ \lim_{n\to\infty}\mathbf P(\sum_{j=1}^{k_n}\xi_{nj}-A_n<x)= (2\pi)^{-1/2}\int_{-\infty}^x e^{-u^2/2} du $$
for some constants $A_n$, $n=1,2,…$, and
$$ \lim_{n\to\infty}\mathbf M|\sum_{j=1}^{k_n}\xi_{nj}-A_n|^{2q}= (2\pi)^{-1/2}\int_{-\infty}^{\infty}|u|^{2q} e^{-u^2/2} du $$
for some $q>0$ if and only if
$$ \lim_{n\to\infty}\mathbf M|\sum_{j=1}^{k_n}(\xi_{nj}- \mathbf M\{\xi_{nj}||\xi_{nj}|<1\})^2-1|^q=0. $$