This is a bit of a silly question, but I need a way to reduce $o_\text{P}(f) + o(g)$ to one term. I know that $o(f) + o(g) = o(\max(f, g))$, but I am not sure how to extend the property when one of those terms is $o_\text{P}$ instead, and I can't seem to find a good answer online.
*Edit: Ahh, forgot to define $o_\text{P}$. I say $X_{n} = o_\text{P}(c_{n})$ for some constant $c_{n}$ (indexed by $n$) if, for all $\epsilon > 0$, $$ \lim_{n \to \infty}\Pr\left(\left|\frac{X_{n}}{c_{n}}\right| \geq \epsilon \right) = 0. $$ This definition is rather controversial. A more popular definition: for all $\delta, \epsilon > 0$, there exists some $n_{0}$ such that, for all $n > n_{0}$, $$\Pr\left(\left|\frac{X_{n}}{c_{n}}\right| > \delta \right) \leq \epsilon. $$ So it's essentially an extension of $o()$ to random variables; thus, if $X_{n} = o_\text{P}(c_{n})$ and $Y_{n} = o_\text{P}(d_{n})$, then $X_{n} + Y_{n} = o_\text{P}(\max(c_{n}, d_{n}))$.
