Suppose $Z\_{1}, Z\_{2}, \dots, $ are independent and identically distributed random variables with mean 0 and variance 1. Put $X\_{nk}=\sigma\_{nk} Z\_{k}$ for $n=1,2, \dots$ and $k=1, 2, \dots, r\_{n}$, where $r\_{n} \to \infty$ as $n \to \infty$ and $\sum\_{k=1}^{r\_{n}} \sigma^2\_{nk}=1$. If $\max\_{1\le k \le r\_{n}} \sigma\_{nk}=o(1)$ as $n\to \infty$, does the Lindeberg condition holds? That is, for any $\eta>0$, do we have 
$$\sum\_{k=1}^{r\_{n}} E[|X\_{nk}|^2 I\_{\{|X\_{nk}| > \eta\}}] \to 0$$ as $n\to \infty$? Thank you for your reply.



The Lindeberg Condition
$\newcommand{\E}{\mathbf{E}}\renewcommand{\P}{\mathbf{P}}\DeclareMathOperator{\var}{Var}$
Yes. If $s\_n=\max\_{1\le k \le r\_n} \sigma\_{nk}$ then we have
$$\E\big[|X\_{nk}|^2 I\_{|X\_{nk}|>\eta}\big] \le \sigma\_{nk}^2 \E\big[|Z\_k|^2 I\_{|Z\_k|>\eta/s\_n}\big]$$
hence
$$\sum\_{k=1}^{r\_n}\E\big[|X\_{nk}|^2 I\_{|X\_{nk}|>\eta}\big] \le \E\big[|Z\_k|^2 I\_{|Z\_k|>\eta/s\_n}\big]$$
and the RHS goes to 0 on account of the existence of the variance.


