I'm confused on the relation between stochastic big O notation and signs. In order to illustrate my question:
(1) consider a sample of i.i.d. real-valued random variables $\{X_i\}_{i}^n$, each with support $\mathcal{X}\subset \mathbb{R}$;
(2) for a fix realisation of $X_i$, $x\in \mathcal{X}$, consider the map $m(\theta, x): \Theta\subset \mathbb{R}\rightarrow \mathbb{R}$.
(A) Suppose you are said that $\theta^*$ maximizes, with respect to $\theta$, $\frac{1}{n}\sum_{i=1}^{n}m(\theta, X_i)$ up to a variable $R_n \in O_p(r_n)$ where $r_n$ is a positive increasing function of $n$.
What should be the interpretation of the last sentence?
My attempt: $\frac{1}{n}\sum_{i=1}^{n}m(\theta^*, X_i)= \sup_{\theta \in \Theta} \frac{1}{n}\sum_{i=1}^{n}m(\theta, X_i)-R_n$ where $R_n=r_n Y_n$, $Y_n\in O_p(1)$ and $Y_n\geq 0$ (so that $R_n\geq 0$).
My doubt is about the sign of $R_n$. Is it correct to believe that $R_n$ should be positive? Does it make sense to assign signs to stochastic big O's?
(B) Suppose also that $\frac{1}{n}\sum_{i=1}^{n}m(\theta^*, X_i)-\frac{1}{n}\sum_{i=1}^{n}m(\tilde{\theta}, X_i)\geq -O_p(r_n)$, where $\tilde{\theta} \in \Theta$
What does it mean that a function is bigger than a $-O_p(r_n)$? What is the meaning of the minus in front of $O_p(r_n)$?