Show that there exist $ b \in (0, \infty)$ such that $\lim_{n \to \infty} exp(bn) \cdot P((1/n)\cdot \sum_{i=1}^n X_i >a)=0$

Question

Let ($X_i$) be an i.i.d. sequence of random variables $X_i: \Omega \to [0, 1], i \in \Bbb N$, on a probability space $(\Omega, F, P)$. Let $a \in (E(X_1), \infty)$. How to show that there exist $ b \in (0, \infty)$ such that $\lim_{n \to \infty} exp(bn) \cdot P((1/n)\cdot \sum_{i=1}^n X_i >a)=0$? I think it can be solved by Chernoff bound, but I can not work it out. Please help, thanks.

Answer 1

Let $t=a-E[X_1]$. Then, notice that $$ P\left( \frac{1}{n} \sum_{i=1}^n X_i > a \right)= P\left( \frac{1}{n} \sum_{i=1}^n (X_i-E[X_1]) > t \right) \le e^{-2nt^2} $$ by Hoeffding's inequality.