Suppose I have two random variables $X\_1$ and $X\_2$. $X\_1,X\_2$ are both sums of random variables, and I can find Chernoff bounds for both variables independently. That is to say, I have
$$Pr[\mid X\_i - \mathbb{E}X\_i \mid \geq \delta \mathbb{E}X\_i] \leq 2\exp(-\frac{\delta^2 \mathbb{E}X\_i}{3}) $$ for both $i=1,2$.


Now, I have a new variable $$Y = \frac{X\_1 - X\_2}{X\_1 + X\_2}$$ I know how I can get concentration bounds using the union bound for both the numerator and denominator. But is there a way that I can get some kind of bound for the fraction itself? Here is what I attempted.


First, get the bounds for the numerator. Then holding the lower and upper bounds of the numerator constant, try to get concentration bounds for the (lower(or upper) bound/denominator random variable) term, i.e., get lower bounds for $\frac{LB(X\_1-X\_2)}{X\_1+X\_2}$, where LB is taking the lower bound we get from the numerators concentration bound. Then try to find probability and lower bound for this new $\frac{constant}{X\_1+X\_2}$ type term. Similarly try to find probability and upper bound using the numerator upper bound term. But the calculation got messy and I couldn't finish it.


Is there a method for solving problems like this?



Chernoff style concentration bound for ratio of variables
$\newcommand{\de}{\delta}$You have the bounds
\begin{equation}
 P(|X\_i-m\_i|\ge \de\_i m\_i]\le2e^{-\de\_i^2 m\_i/3}
\end{equation}
for $i=1,2$ and $\de\_i>0$, where
\begin{equation}
 m\_i:=EX\_i. 
\end{equation}
Clearly, these bounds are useful only if $m\_i>0$ for $i=1,2$, which will be henceforth assumed. Assume also that $0<\de\_i<1$ for $i=1,2$.


Then on the event
\begin{equation}
\begin{aligned}
 A&:=\{|X\_1-m\_1|<\de\_1 m\_1,|X\_1-m\_1|<\de\_2 m\_2\} \\ 
 &=\{m\_1(1-\de\_1)<X\_1<m\_1(1+\de\_1),m\_2(1-\de\_2)<X\_2<m\_2(1+\de\_2)\}
\end{aligned}
\end{equation}
we have $X\_1>0$ and $X\_2>0$, which implies that
\begin{equation}
 Y = \frac{X\_1-X\_2}{X\_1+X\_2}
\end{equation}
is increasing in $X\_1$ and decreasing in $X\_2$, so that the event  

\begin{equation}
B:=\Big\{\frac{m\_1(1-\de\_1)-m\_2(1+\de\_2)}{m\_1(1+\de\_1)+m\_2(1-\de\_2)} <Y<\frac{m\_1(1+\de\_1)-m\_2(1-\de\_2)}{m\_1(1-\de\_1)+m\_2(1+\de\_2)}\Big\}
\end{equation}
occurs. That is, $A\subseteq B$ and hence $1-P(B)\le1-P(A)$.


So, by the union bound, we get the concentration result:
\begin{equation}
 1-P(B)\le 2e^{-\de\_1^2 m\_1/3}+2e^{-\de\_2^2 m\_2/3}. 
\end{equation}
One can now (quasi-)optimize the choices of $\de\_1$ and $\de\_2$, depending on one's objectives.


