Let $E\_1, \dots, E\_N$ be independent events, each of probability $p$, where $p$ is very close to $0$. Let $A\_N = \frac{1}{N} ( 1\_{E\_1} + \dots + 1\_{E\_N} )$ be the proportion of the events $E\_i$ that occur. We expect $A\_N$ to be tightly concentrated around its mean $p$.


Suppose we want to estimate something like $\mathbb{P}(A\_N > p^{1/2})$. On the one hand, the multiplicative difference $p^{1/2}/p$ is huge, but on the other hand the additive difference $p^{1/2} - p$ is very small. All the standard concentration inequalities (Azuma-Hoeffding, Chernov, etc.) give an upper bound for the above probability in terms of the additive difference, which gives only a very slow exponential decay rate as $N \to \infty$ for the above probability if $p$ is very small.


My question is: which phenomenon is closer to the truth? Should the event $\{A\_N > p^{1/2}\}$ be very rare because $p^{1/2}/p$ is huge, or should it be not so rare because $p^{1/2} - p$ is tiny? If the former, are there any references out there for concentration inequalities that capture that?



Concentration inequalities for very rare events on a multiplicative scale
Let $n:=N$. Let us show that for all natural $n$ and all $p\in(0,1)$
$$P(A\_n>\sqrt p)\le\frac{\sqrt p+p}{1+p},\tag{1}$$
so that $P(A\_n>\sqrt p)\to0$ whenever $p\downarrow0$.


Consider first the case when $n\ge1/\sqrt p$, so that $1/n\le\sqrt p$.
In view of [Cantelli's inequality](https://en.wikipedia.org/wiki/Cantelli%27s_inequality),
$$\begin{aligned}
P(A\_n>\sqrt p)&\le\frac{p(1-p)/n}{p(1-p)/n+(\sqrt p-p)^2} \\ 
&\le\frac{p(1-p)\sqrt p}{p(1-p)\sqrt p+(\sqrt p-p)^2} \\ 
&=\frac{\sqrt p+p}{1+p}, 
\end{aligned}
$$
so that (1) holds if $n\ge1/\sqrt p$.


In the remaining case, when $n<1/\sqrt p$, we have $\sqrt p<1/n$ and hence
$$P(A\_n>\sqrt p)=P(A\_n>0)=1-(1-p)^n\le np<\sqrt p<\frac{\sqrt p+p}{1+p},$$
so that (1) again holds.


Thus indeed, (1) holds for all natural $n$ and all $p\in(0,1)$. (It actually holds for all $p\in[0,1]$.)


