Let $\{U_n\}$ be a sequence of i.i.d Uniform$(0,1)$ random variables and $\{a_k\}$ be a sequence of i.i.d random variables such that, $P(a_k = \pm1) = \frac{1}{2}$ and are independent of the $\{U_n\}$. Define the quantities,
$$X_n = \sum\limits_{k=1}^nU_ka_k, \;\;\;\; Y_n = \sum\limits_{k=1}^n(U_k^2 - 1/2)a_k$$
Find the limit, $\lim\limits_{n\rightarrow\infty}P\left(|X_n| > Y_n\sqrt{\frac{20}{7}}\right)$.
We can easily calculate that $$E[U_ka_k] = 0, \; E[U_k^2] = \frac{1}{3}$$ and similarly, $$E[(U_k^2-1/2)a_k] = 0, \; E[(U_k^2-1/2)^2] = \frac{7}{60}$$ So by the Central Limit Theorem, we get,
$$ \frac{1}{\sqrt{n}}X_n \xrightarrow{D} N(0,1/3), \;\;\;\; \frac{1}{\sqrt{n}}Y_n \xrightarrow{D} N(0,7/60)$$
Ideally, I can use something like the Continuous Mapping Theorem to deduce the limiting distribution of $\dfrac{|X_n|}{Y_n}$, but the question makes no additional assumptions on the joint distribution of $(X_n, Y_n)$. Is there another approach I can take from here?
EDIT: I went ahead and calculated the covariance of $U_ka_k$ and $(U_k^2-1/2)a_k$,
$$ E[U_ka_k(U_k^2-1/2)a_k] = E[U_k(U_k^2-1/2)] =\int_0^1(u^3-1/2u)du = 0$$
since $a_k^2 = 1$ with probability $1$. Now we can apply the multivariate CLT like @LandonCarter says.