# random variable, random sum

I have problem to show the following. Maybe someone has an idea :/

Let $$X_1,...,X_n$$ be i.i.d. copies of a random variable X with $$|X|<1$$. Let $$S_n=X_1+..+X_n$$. Then for any $$A>0$$: $$\mathbb{P}(|S_n-n\mathbb{E}[X]|\geq An)\leq C_A e^{-c_An}$$ for some constants $$C_A,c_A>0$$ depending on $$A$$.

I have tried this before:

$$\mathbb{P}(|S_n-n\mathbb{E}[X]|\geq An)= \mathbb{P}(|\sum_{i=1}^nX_i-\mathbb{E}[X]|\geq An)= \mathbb{P}(|\frac{1}{n}\sum_{i=1}^n X_i-\mathbb{E}[X]|\geq A)\leq \mathbb{P}(\frac{1}{n}\sum_{i=1}^n| X_i-\mathbb{E}[X]|\geq A)$$

• What have you tried so far? – Mark May 4 at 9:41
• I have tried this before:) (see above) – toni_iva May 4 at 9:48
• I have adjusted my answer, based on your try. Good luck! – Mark May 5 at 9:24

Hint: Markov's inequality

(Extended version for monotonically increasing functions)