Let $0 < a_1 < a_2 < · · ·$ be fixed real numbers and let $\{X_k\}_{k \geq 1}$ be a sequence of i.i.d. random variables with zero mean and unit variance. Let $T_n = \sum_{i=1}^{n} a_iX_i$. Find a simple sufficient (deterministic) condition on the sequence $\{a_k\}_{k \geq 1}$ to ensure that $\frac{T_n}{\sqrt{Var(T_n)}}$ converges in distribution to $\mathcal{N}(0, 1)$. Use your sufficient condition to verify the asymptotic normality of $T_n$ when $a_k = k^\alpha$ where $\alpha > 0$ is a fixed real number.

Here is my approach: I think I need to determine conditions on $a_i$ such that the Lindeberg condition is satisfied. $E[T_n] = 0$ and $Var(T_n) = \sum_{i = 1}^{n}a_{i}^{2}$. Define $X_{n,k} = \frac{a_kX_{k}}{\sqrt{Var(T_n)}}, 1 \leq k \leq n.$ Clearly, $\{X_{n,k}\}_{1\leq k \leq n}$ are independent, $E[X_{n,k}] = 0$ and $\sum_{k = 1}^{n}E[X_{n,k}^{2}] = 1 < \infty$. Let $\epsilon > 0$.

$\sum_{k=1}^{n} E[X_{n,k}^{2}]{1}_{\{|X_{n,k}|>\epsilon\}} = \frac{1}{Var(T_n)}\sum_{k=1}^{n}E[a_k^2X_k^21_{\{|X_k| > \epsilon \frac{ \sqrt{Var(T_n)}}{a_k}\}}] \leq \frac{na_n^2}{Var(T_n)} E[X_1^21_{\{|X_1| > \epsilon \frac{ \sqrt{Var(T_n)}}{a_k}\}}].$

I want the RHS of the above to approach $0$ as $n \rightarrow \infty$ for the Lindeberg condition to hold. I do not know how to proceed from here.


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