# Concentration inequalities for product of gaussians

Are there any concentration inequalities (i.e. probability bounds on how a random variable deviates from its expectation) for the product of $$n$$ independent gaussian random variables with zero means and equal variances? What about different means and variances?

• do you assume the independence of gaussians or no? Nov 29 '19 at 12:13
• @pointguard0 yes Dec 2 '19 at 7:05

The theorem states that for independent standard normal Gaussians random variables $$g_i$$ one has $$\mathbb{P}(X \le t) = \mathbb{P}(X \ge -t) = f_{0.5, 1}\bigg(\frac{2^n} t\bigg),$$ where $$f_{\nu, \xi}(u) = \nu + \frac 1 {2^{\xi}} \cdot \frac 1 {\pi^{n/2}} \sum_{k=0}^\infty u^{-0.5 - k} \sum_{j=0}^{n-1} H_{kj} \cdot [\log u]^j,$$ for details and definition of $$H_{kj}$$ see the paper itself.