Sum of independent random variables having exponential tails

Question

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.

Answer 1

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.