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

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.

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.