I am now working on an exercise problem (Ex. 2.29) from Foundations of Data Science by A. Blum, J, Hopcroft, and R. Kannan:
Let $x$ and $y$ be $d$-dimensional zero mean, unit variance Gaussian vectors. Prove that $x$ and $y$ are almost orthogonal by considering their dot product.
Below is my attempt:
Inspired by this post, I let $z = x^Ty/|x|$ and concluded that $z\sim\mathcal{N}(0,1)$ and that $z$ is independent of $x$, by rotational invariance of spherical Gaussian. Then, for $\epsilon > 0$, I tried to directly compute $\mathbb{P}(|x^Ty| > \epsilon)$ by conditioning on $|x|$, i.e., $$ \mathbb{P}(|x^Ty| > \epsilon) = \int_0^{\infty} \mathbb{P}(|z| > \epsilon/r) f(r)\mathrm{d} r = \int_0^{\infty} 2[1 - \Phi(\epsilon/r)]f(r)\mathrm{d}r, $$ where $f(r)$ denotes the PDF of $|x|$.
This is what I have done so far. Yet I am not sure how to proceed. I know that by the Gaussian annulus theorem, most of $|x|$ will be concentrated around the surface $\sqrt{d}\mathbb{S}^{d-1}$, but how am I supposed to bound $f(r)$ and the whole integral above? Any help will be greatly appreciated!