1
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ Research level question? $\endgroup$ – Did May 29 '13 at 6:00
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.