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Let $X_n$ and $Y_n$ be random variables such that $X_n=o_p(1)$, $Y_n=o_p(1)$, $X_n - Y_n = o_p(1)$. Is the following correct?

$o_p(X_n) + o_p(Y_n) = o_p(|X_n - Y_n|)$

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I don't think this even holds for non-stochastic sequences. But I could be wrong. My maths is getting rusty.

For example, let $X_n = \frac{1}{n}$ and $Y_n = -\frac{1}{n}$. Definitely they both go to zero in probability. Recall that $o_p\left(X_n\right)$ stands for some sequence $A_n$ such that $A_n/ X_n \overset{p}\to 0$. So if we let $o_p\left(X_n\right)$ and $o_p\left(Y_n\right)$ both be $\frac{1}{n^2}$. It is definitely not true that $$\frac{\frac{1}{n^2} + \frac{1}{n^2}}{\frac{1}{n} - \frac{1}{n}} \to 0$$

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  • $\begingroup$ Hi, I just editted the question to be more appropriate. $\endgroup$ – Hercules Jan 11 at 14:56
  • $\begingroup$ @Hercules I think if you let $Y_n = 1/n$, you still get a counterexample. $\endgroup$ – TJnFvYLDu3 Jan 11 at 17:14
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Not exactly an answer to your question, but the following holds true:

If $T_n=o_p(n^\lambda)$ and $S_n=o_p(n^\mu)$ then $T_n+S_n=o_p(n^{\max[\lambda,\mu]})$, see e.g. White (1984) Asymptotic Theory for Econometricians, pp.25-6.

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