Let $A_1,\dots,A_n\in \mathbb{R}^{d\times d}$ be some matrices. Suppose we sample $x_1,\dots,x_n,y\sim \mathcal{U}(\mathbb{S}^{d-1})$, where $\mathcal{U}(\mathbb{S}^{d-1})$ is the uniform distribution over the unit sphere. I want to show that with high probability over the samples we have:

$ \left\langle \frac{A_1x_1 + \dots A_nx_n}{||A_1x_1 + \dots A_nx_n||}, y \right\rangle \leq c \cdot \frac{1}{\sqrt{d}} $

where $c$ is some constant.

Now, for one vector and one matrix, I can show it using standard concentration arguments with $c=2$ and get that the correlation is small with probability exponentially close to $1$.

I don't know how to generalize it to a general number of matrices. My intuition is that it is still true since it looks impossible to pick matrices $A_i$ which after the normalization will have a high correlation for all choices of the $x_i$'s and $y$.

As a side note, I'm also interested in the case that all the distributions are Gaussian $\mathcal{N}\left(0,\frac{1}{d}I\right)$ instead of uniform over the unit sphere - but we still normalize over the sum.