TITLE: Asymptotic Behavior of Ratio of Random Variables
QUESTION [7 upvotes]: Let $X_n \ge 0$ and $Y_n > 0$ be sequences of random variables. Suppose that for a positive sequence of numbers $\{a_n\}$, $P(Y_n < a_n) \rightarrow 0$ as $n \rightarrow \infty$ and $X_n/a_n \rightarrow 0$ in probability.
My question is: can we say that $X_n/Y_n \rightarrow 0$ in some  (probabilistic) sense? If there are further conditions under which this is true, what are they?
I know that if $X_n$ and $Y_n$ converged to constants $a$, $b$ in probability, we would have $X_n/Y_n \rightarrow a/b$ in probability. 
Some things I have tried:

Split the ratio in two cases with indicators $1(Y_n \ge a_n)$ and $1(Y_n \le a_n)$.
Try to combine Markov with Cauchy-Schwarz to provide an upper bound.

Thanks in advance!

REPLY [3 votes]: $\frac{X_n}{a_n}=o_p(1)$, $\frac {a_n}{Y_n}=O_p(1)$, hence their product $\frac{X_n}{Y_n}$ is $o_p(1)$.