I am working on an exercise stated as follows:
Assuming $\sum_{k=1}^{\infty}p_{k}=\infty$, give an example of independent random variables $(X_{k})$ such that $Var(X_{k})\leq p_{k}<\infty$ and $\mathbb{E}X_{k}=0$ such that $\limsup_{n}X_{n}=1$ almost surely and $n^{-1}\sum_{k=1}^{n}kX_{k}=\infty$.
The first example coming to my mind is just to take $X_{k}=1$ for all $k$, so we have the desired divergent sum and $\limsup X_{n}=1$, and $Var(X_{k})=0$, but firstly $\mathbb{E}X_{k}=1$, not $0$ and I don't know if they are independent or not....
I have no other idea for now... I can take something like $$\mathbb{P}(X_{k}=-1)=\mathbb{P}(X_{k}=1)=1/2,$$ but then it is really hard to compute the infinite sum and $\limsup$...
Any idea? Thank you!