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.

$\endgroup$
1
  • 3
    $\begingroup$ Research level question? $\endgroup$
    – Did
    Commented May 29, 2013 at 6:00

1 Answer 1

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 and acknowledge you have read our privacy policy.

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