Given a non-zero high-dimensional vector, $v\in (\mathbb{R} \setminus \{0\}) ^ d$, and a random sign vector $s \in \{-1,1\}^d$ (i.e., each entry is a rademacher random variable).


Empirically, I find that the distribution of $s \cdot v$ seems to be $\mathcal{N}(0, \frac{||v||\_2^2}{d})$, or at least I can' find an example when it isn't as $d \to \infty $.


(If I would have added that the coordinates are independent and iid, this would be just the Central Limit Theorem.)


Is there some known form of CLT that explains why this would work for a general vector multiplied by random signs?
(or a counterexample that I'm missing)


Thank you!



The distribution of the sum of a non-zero vector with random signs
Of course, your empirically motivated conjecture will not hold in general. E.g., it will not hold if $v\_1^2$ is much greater than $\sum\_{i\ge2}v\_i^2$, where the $v\_i$'s are the coordinates of $v$.


On the other hand, by (say) the [Berry--Esseen inequality](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem#Non-identically_distributed_summands), the dot product $s\cdot v$ of $s$ and $v$ will be approximately normal if $\sum\_i|v\_i|^3$ is much less than $(\sum\_i v\_i^2)^{3/2}$, which will in turn follow if e.g. $\max\_i v\_i^2$ is much less than $\sum\_i v\_i^2$. (I guess the condition that $\sum\_i|v\_i|^3$ is much less than $(\sum\_i v\_i^2)^{3/2}$ held in your empirical studies.)


