Let ${x_1,y_1,x_2,y_2,...}$ be a be a sequence of mutually independent random variables.
$x_i=1$ with probability $p_i$ and $x_i=0$ with probability $1-p_i$.
$y_i=1$ with probability $q_i$ and $y_i=0$ with probability $1-q_i$.
Assume that there exists $m>0$ such that $\frac{1}{m}<\frac{p_i}{q_i}<m$ for all $i$.
Let $\bar{x}_n=\frac{1}{n}\sum_{1}^{n}x_i$ and $\bar{y}_n=\frac{1}{n}\sum_{1}^{n}y_i$.
How to approximate $\frac{P(\bar{x}_n=k)}{P(\bar{y}_n=k)}$ when $n$ is large for a given $k\in(0,1)$ if we know $E(\bar{x}_n)$ and $E(\bar{y}_n)$?
Thanks
Any related questions/topics/theorems are also welcome.