This is problem is an intermediate step that I encounter in my theoretical computer science study, to be specific, precision sampling.
Say we have a bunch of independent random variables $u_i \sim \exp(1)$ for $i = 0, 1, ..., m$ and some determined positive constants $a_1, a_2, ...a_m$. I wanna show that exist $k = \Omega(1/\log m)$ such that $$ \Pr_{u_i}\Big[\sum_i\frac{a_i}{u_0} > k\sum_i\frac{a_i}{u_i} \Big] > 90\% $$
Intuitively, this problem is should be super easy: the same number is more likely to divided by a smaller number on the LHS, not to mention there are $m$ copies of them. However, to prove it rigorously, I cannot get anything out. I do not see a general way of handling all the sum of inverses. Aside, we also have a problem that the expectation of inverse exponential distribution blows out to infinity.
Any help is appreciated.