Suppose $X_1, X_2,\dots, X_n$ are iid random variable,$P(X_i=-\infty)>0$,$P(X_i>v)< e^{-v}\forall v>0$, $X$ 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.

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.