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Let $\xi_i$ with $i\in\{1,\dots,n\}$ be iid random variables and let $Q(x,y)$ be a rational function.

I need to compute one $x$ that satisfies $$\frac{1}{n}\sum_{i=1}^n Q(x,\xi_i)=0.$$ This is a horrible equation for big $n$ because each denominator is different, but $\xi_i$'s have small variance, so there seems to exist a root close to each root of $Q(x,E[\xi_i])$.

Is there any way to approximate this mean? Is there some kind of expansion or some limit theorem that can be used to get an approximation of a root of this equation?

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