I've been having some trouble with the following question:
Let $X_1,X_2,...$ be almost surely positive random variables converging in distribution. Can you always find reals $(c_n)_{n\geq 1}$ such that $\frac{X_n}{c_n}$ converges in distribution to an almost surely positive random variable?
Just to be clear, by positive here I mean strictly bigger than $0$. In a previous question I was asked to find a sequence of reals for an arbitrary sequence of random variables such that the above ratio converges almost surely to 0 and I was able to do this, but I am not sure how to apply it to this question (if it even can be applied). My first reflex was to think that one cannot always find such reals but I've been struggling to find the right sequence of variables for a counterexample.
If $X_n$ and $\frac {X_n} {c_n}$ both converge to non-constant random variables then $(c_n)$ must converge. This is proved in Brieman's book on Probability. (This is çalled the 'Convregence of Types Theorem'). From this you get an easy counter-example by making the limit of $(X_n)$ not identically $0$ but taking the value $0$ with positive probability. [ I will leave this elementary construction to you].