TITLE: Concentration inequalities for very rare events on a multiplicative scale
QUESTION [9 upvotes]: 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?

REPLY [7 votes]: 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,
$$\begin{aligned}
P(A_n>\sqrt p)&\le\frac{p(1-p)/n}{p(1-p)/n+(\sqrt p-p)^2} \\ 
&\le\frac{p(1-p)\sqrt p}{p(1-p)\sqrt p+(\sqrt p-p)^2} \\  
&=\frac{\sqrt p+p}{1+p}, 
\end{aligned}
$$
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 $p\in[0,1]$.)