Thank you.. I think I see now. On the other hand, can we say something about the behavior of $\arg\max_x \left\{qF(n,x)+(1−q)G(n,x)\right\}$ as $n\rightarrow \infty$?

I am sorry I couldn't follow your argument although I think your approach would be of great help. For instance when I rewrite I get $P(AY_n+B(n-Y_n)+Cx+Z\sigma<n)$ but this looks something different. Also the argument where you refer to the law of large numbers is not clear to me. Could you clarify a bit more?

@IosifPinelis $A,B,C\in\mathbb R_{++}$ fixed and $p\in(0,1)$.

@behradmahboobi I mean growth in order of $n^2$. It is not that necessary, in fact, order can be higher than 2.

@PerAlexandersson I tried to specify a bit more by adding an update.