# Tail inequality on sum of product of normal variables

For independent random variables $x_1,..,x_n$ and $y_1,...,y_n$ following normal distribution $N(0,1)$, I need a simple estimate formula for $P(| \sum_1^n x_iy_i | \leq nt ) \leq e^{(?)}$ for $t>1$. Thanks.

• Are the $x_i$ and $y_i$ independent? If so, the inner product is a subexponential random variable in the sense of R. Vershynin. See his recent expository introduction to nonasymptotic random matrix theory for relevant bounds of the type you desire. – cardinal Oct 20 '12 at 9:58
• Also, do you mean $\geq$ instead of $\leq$ inside the probability function? – cardinal Oct 20 '12 at 10:00
• Thanks. You are right, it should be $P(|\sum_1^n x_iy_i| \geq nt)≤e^{(?)}$. Can you be more specific about the reference? – user16093 Oct 20 '12 at 11:55
• Yes, all $x_i$ and $y_i$ are independent from each other. – user16093 Oct 20 '12 at 13:13
• A very similar question was asked on MO: mathoverflow.net/questions/109989/… I'm not totally satisfied with any of the answers. I think you can do better. See stats.stackexchange.com/questions/4816/… . – Douglas Zare Oct 21 '12 at 7:28