The Lindeberg Condition

Question

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.

Answer 1

$\newcommand{\E}{\mathbf{E}}\renewcommand{\P}{\mathbf{P}}\DeclareMathOperator{\var}{Var}$ 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.