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I have a question related to stochastic little o properties. Let $\{X_n\}_n$ be a sequence of real-valued random variables defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $$X_n\in o_p(\frac{1}{n})$$

$(*)$ Does $X_n\in o_p(\frac{1}{n})$ imply that $X_n\in o_p(1)$?

My attempt is : yes, since $\frac{1}{n}=o(1)$ $\rightarrow$ $\frac{1}{n}=o_p(1)$; write $X_n=\frac{1}{n}o_p(1)=o_p(1)o_p(1)=o_p(1)$

Keeping in mind the answer to $(*)$, suppose now I have two sequences of real-valued random variables defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $\{Y_n\}_n$, $\{Z_n\}_n$ such that $$(**) Y_n\geq Z_n -o_p(\frac{1}{n})$$

In view of the (potentially) positive answer to $(*)$ can I then say that $(**) $ implies

$$ (***) Y_n\geq Z_n -o_p(1) $$

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