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I was trying approximate the variance of a ratio of two random variables. I used to approximate it through Taylor's expansion:

Assume $\sqrt{n}\big(X-E(X)\big)=O_p(1)$, $\sqrt{n}\big(Y-E(Y)\big)=O_p(1)$, then

$\sqrt{n}\big(f(X,Y)-f(E(X),E(Y))\big)=f_x\sqrt{n}(X-E(X))+f_y\sqrt{n}(Y-E(Y))+o_p(1)$

and in the calculation after this the $o_p(1)$ changes into $o(1)$ after taking variance, and can be neglected in the approximation -- At least that's what my lecture notes said.

But today I suddenly had this doubt: why can I just take $Var(o_p(1))$ to be $o(1)$, or even, why can I take $E(o_p(1))=o(1)$? Since p-convergence to zero not necessarily imply variance or expectation also converges to zero, I am no longer sure why I can do the above approximation in that particular way.

Is the approximation I did above correct? If not, what is the correct way to approximate the variance of the ratio?

What are some technical conditions that can guarantee that the above approximation is true?

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