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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.

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  • $\begingroup$ Unfortunately this will not hold in general - suppose $X_i,Y_j \sim \mathcal{N}(0,1)$. Then both $X$ and $Y$ are highly concentrated, but the ratio $X/Y \overset{\mathrm{Law}}{=}\tan \Theta, \Theta \sim \mathrm{Unif}[0,2\pi)$ for every $N_1 = N_2$ (quick exercise: prove this), which is rather unconcentrated. $\endgroup$ Dec 20, 2018 at 3:39
  • $\begingroup$ I don't think the above comment is very useful b/c it implies there's nothing to be done in these scenarios. But in fact this is a pretty well studied problem at least asymptotically and/or with a normality assumption. See, for instance, Fieller's theorem, the delta method, and the following paper: arxiv.org/pdf/0710.2024.pdf I too am interested in what assumptions can be relaxed in terms of asymptotics and/or normality and to see if you can get concentration for a fixed $N_1, N_2$; i.e., if you can get a concentration inequality. Did you make any progress with this? $\endgroup$
    – Adam
    Feb 20, 2019 at 1:30
  • $\begingroup$ One possibility is if you can bound the denominator away from zero in which case concentration inequalities which apply to lipschitz functions of iid random variables may be of use? $\endgroup$
    – Adam
    Feb 20, 2019 at 1:32

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