Say we have the following inequality: $ A < B < C$ where $A, B$ and $C$ are positive integer-valued random variables. Assume that $A$ is concentrated in $O(m)$ values with high probability and $C$ is concentrated in $O(n)$ values whp.
Then, does it follow that B is concentrated in $O(m) + O(n) + \mathbb{E}[C-A]$ values with high probability? Can we improve this?
Thanks!