Is there a sequence of distributions $(F_n)$ with $F_n(0)=0$ such that $\frac{E[\max(X_{1,n},X_{2,n})]}{E[X_{1,n}]} \to +\infty $ as $n\to +\infty$, where $X_{1,n}$ and $X_{2,n}$ are independently and identically drawn according to $F_n$?
It would be great if we can find a "natural" $F_n$ that we are familiar with.
I would like to use it to prove some result in finance field.