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



Probability inequalities
This is a standard exercise in large deviations. The exponential rate of decay for the large deviations of sums of i.i.d. random variables can be derived using Cramer's Theorem (see section 2.2 in Dembo and Zeitouni's book). 


In the statement of the problem above, Cramer's theorem gives that 
$$P(|X| > \epsilon ) = e^{ - N I(\epsilon) + o(N) }, $$
where $I(x)$ is the large deviation rate function. In this example (unless I screwed up in my calculation)
$$ I(x) = \frac{1}{2}\left( \sqrt{1+4x^2}-1+\log(\sqrt{1+4x^2}-1)-\log(2x^2) \right) .$$


