Suppose $X\_1, X\_2,\dots, X\_n$ are iid random variable,$P(X\_i=-\infty)$ is allowed,$P(X\_i>v)< e^{-v}\forall v>0$, $X$ is distributed as $X\_i$, if $c$ is a finite real number such that $E(X)<c$, then show that there is $A>0, r<1$ such that $P(X\_1+\dots+X\_n>nc)< Ar^n\forall n$. Can apply Chernoff or Hoefding bound here, to apply do I need to know what distribution $X\_i$ is following? I am a bit confused. what is the role of $X$ here? Thanks for helping.



Sum of independent random variables having exponential tails
By the Markov--Bernstein inequality (incorrectly referred to as Chernoff's)
\begin{equation}
 P(X\_1+\dots+X\_n\ge nc)\le e^{-tnc}Ee^{t(X\_1+\dots+X\_n)}
 =e^{-tnc}(Ee^{tX})^n=e^{ng(t)} 
\end{equation}
for real $t\ge0$, where 
\begin{equation}
 g(t):=-tc+\ln Ee^{tX}. 
\end{equation}
The condition $P(X\_i>v)< e^{-v}\ \forall v>0$ implies $Ee^{tX}<\infty$ and hence $\frac d{dt}\,Ee^{tX}=EXe^{tX}\in\mathbb R$ for real $t<1$. So, $g(0)=0$ and $g'(0)=-c+EX<0$, whence $g(s)<0$ for some real $s>0$. We conclude that 
\begin{equation}
 P(X\_1+\dots+X\_n\ge nc)\le r^n, 
\end{equation}
where $r:=e^{g(s)}\in(0,1)$, as was desired. 


