Correlation for a Sum of random vectors from the sphere multiplied by matrices

Question

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.

Answer 1

Is $Y$ independent of $(X_1,\ldots,X_n)$? If yes, write it.

If yes, it suffices to prove that for any fixed unit vector $u$, $\langle u,Y \rangle \le c/\sqrt{d}$ with high probability. By rotational invariance, one may assume that $u$ is $e_d$, the last vector of the canonical basis of $\mathbb {R}^d$.

Moreover, the random variable $Y$ can be constructed as $Z/|Z|$, where $Z \leadsto \mathcal{N}(0,O_d)$. Then $$\langle u,Y \rangle \le c/\sqrt{d} \iff Z_d \le (c/\sqrt{d})|Z| \iff Z_d^2 \le (c^2/d)(Z_1^2 + \cdots + Z_d^2).$$

Since $Z_1^2 + \cdots Z_{d-1}^2$ and $Z_d^2$ are independent r.v. with distributions Gamma$((d-1)/2,1/2)$ and Gamma$(1/2,1/2)$, the distribution of$Z_d^2/(Z_1^2 + \cdots Z_{d-1}^2)$ is Beta $(1/2,(d-1)/2)$, so the probability above can be computed explicitely.