I have a very general question related to inequalities containing stochastic big o notation.
Introduction: consider two sequences of real-valued random variables $\{X_n\}_n$, $\{Y_n\}_n$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider also a function $g: \mathbb{N}\rightarrow \mathbb{R}$. Suppose $$(*) X_n\geq Y_n-O_p(g(n))$$
We know that $(*)$ is equivalent to $$(**) X_n\geq Y_n+O_p(g(n))$$
Question: take for example Theorem 5.52 at page 75 of van der Vaart "Asymptotic Theory"; among the sufficient conditions we have $\mathbb{P}_nm_{\hat{\theta}_n}\geq \mathbb{P}m_{\theta_0}-O_p(n^{\frac{\alpha}{2\beta-2\alpha}})$. Is there any reason for stating the theorem using as sufficient condition $\mathbb{P}_nm_{\hat{\theta}_n}\geq \mathbb{P}m_{\theta_0}-O_p(n^{\frac{\alpha}{2\beta-2\alpha}})$ instead of $\mathbb{P}_nm_{\hat{\theta}_n}\geq \mathbb{P}m_{\theta_0}+O_p(n^{\frac{\alpha}{2\beta-2\alpha}})$ ?