How can I obtain a concentration bound (concentration inequality) of a random variable $Z$, which is a ratio of $X$ to $Y$, when both $X$ and $Y$ are the sums of IID random variables $X_1,...,X_{N_1}$ and $Y_1,...,Y_{N_2}$, respectively?
$$ \begin{align} Z &= \frac{X}{Y} \\ X = &\frac{1}{N_1}\sum_i^{N_1} X_i ,\;\;\; Y =\frac{1}{N_2} \sum_j^{N_2} Y_j \end{align} $$ or, does $Z$ concentrate in the first place?
I'm concerning this problem because my goal is to take the expectation of a Taylor expansion with respect to $Z$. I want to compute something like
$$ \mathbb{E}[f(Z)]\approx f(\mathbb{E}[Z]) + \frac{1}{2}\text{Var}(Z) f''(\mathbb{E}[Z]) $$
From the StackExchange post (CrossValidated: Expectation of Taylor Series), the above approximation of the expectation is acceptable when the random variable is highly concentrated. I am curious whether the approximation is acceptable even when $Z$ is the quotient of such variables.