Minimizing variance of sequence of independent but not identically distributed random variable

Question

I tried to work on the problem

Let $(X_n)$ be a sequence of independent random variables with $E[X_n]=\mu$ and $Var[X_n]=n$ for every $n \in \mathbb{N}$. Find the statistic of the form $\sum_{i=1}^nw_iX_i$, where $w_1,\cdots,w_n$ are non-negative numbers that add up to 1, that has the smallest possible variance.

I really have no idea how to solve this problem. I know that if $X_n$ are iid, then $\bar{X}_n$ has the smallest variance but in this case $X_n$ are not iid.

Answer 1

Since $X_i$s are independent of each other, $$ \begin{align} Var{\sum_{i=1}^n w_i X_i} &= \sum_{i=1}^n w_i^2 Var X_i \\ &=\sum_{i=1}^n iw_i^2 \end{align} $$ Apply Cauchy's inequality, $$ \sum iw_i^2 \sum \frac1i \ge \left(\sum w_i \right)^2 = 1 $$ which implies $$ Var\sum w_i X_i \ge \frac{1}{\sum\limits_{i=1}^n \frac1i} $$