# Rate of convergence of sum of two random variables

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?

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

## 2 Answers

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 $$\frac{\frac{1}{n^2} + \frac{1}{n^2}}{\frac{1}{n} - \frac{1}{n}} \to 0$$

• Hi, I just editted the question to be more appropriate. – Hercules Jan 11 at 14:56
• @Hercules I think if you let $Y_n = 1/n$, you still get a counterexample. – TJnFvYLDu3 Jan 11 at 17:14

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.