Say I have n independent variables $\{X_1,X_2 \dots X_n\}$ with Expectation 0 such that $Pr(|X_n| > \alpha) < e^{-\lambda \alpha}$. Can we produce chernoff type inequalities for the sum of these random variables ? One idea I have to use the variance of these random variables . Can we use the given property to bound the variance of these random variables ?
Thanks in advance
The hypothesis implies that the random variable $\mathrm e^{\mu|X_k|}$ is integrable for every $\mu\lt\lambda$ and every $k$. Fix some positive $\mu\lt\lambda$. By independence, the random variable $\mathrm e^{\mu(|X_1|+|X_2|+\cdots+|X_n|)}$ is integrable. By the triangular inequality, $\mathrm e^{\mu|X_1+X_2+\cdots+X_n|}\leqslant\mathrm e^{\mu(|X_1|+|X_2|+\cdots+|X_n|)}$ hence the random variable $\mathrm e^{\mu|X_1+X_2+\cdots+X_n|}$ is integrable. In particular, $P(|X_1+X_2+\cdots+X_n|\gt\alpha)\leqslant C_n\cdot\mathrm e^{-\mu\alpha}$ for some finite constant $C_n$.