Hi everyone,

I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts.

My problem is to find an exponential upper bound over the probability that the linear combination of unbounded i.i.d. random variables, which are in fact the multiplication of two i.i.d. Gaussian, exceeds some certain value, i.e., $\mathrm{Pr}[\lvert X \rvert \geq \epsilon] \leq \exp(?)$, where $X = \sum_{i=1}^{N} \alpha_i w_iv_i$, $w_i$ and $v_i$ are generated i.i.d. from $\mathcal{N}(0, \sigma)$, and $0 \leq \alpha_i \leq 1$ is a coefficient.

I tried to use the Chernoff bound using moment generating function (MGF), but the derived bound was not so tight. The main issue in my problem is that the random variables are unbounded, and unfortunately I can not use the bound of Hoeffding inequality.

I will be to happy if you help me find some tight exponential bound . Thanks in advance