Assume that the dot equality notation $a_n \doteq b_n$ denotes equality in the exponential scale for two positive sequences $\left\{ a_n\right\}$ and $\left\{b_n\right\}$, implying that the limit as $n \rightarrow \infty$ of $\frac{1}{n} \log \frac{a_n}{b_n}$ tends to zero.
Now let us have a binomial distribution, with exponentially many terms and exponentially small success probability, i.e. $$N \sim \operatorname{Binomial}\left(e^{n A}, e^{-n B}\right).$$ Additionally, it's known that $A > B$. I aim to demonstrate that $N \doteq e^{n(A-B)}$ with probability approaching one super-exponentially fast. Alternatively, it may be shown that $N \doteq e^{n(A-B)}$ holds almost surely as $n$ approaches infinity. I have raised a related query in in this topic though this question asserts a stronger statement.