Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. To be clear, $A$ is a matrix whose entries are chosen from $\{1,-1\}$.
Let $B = A^T A$. We know, for example, that $B$ is a square and symmetric $n$ by $n$ matrix with all its diagonal entries equal to $m$ exactly. I am trying to learn how to calculate the expected Frobenius and spectral norm of $B$. We can assume both $m$ and $n$ are large if that helps give a good bound or estimate.
How can you calculate $\mathbb{E}(||B||_F)$ and $\mathbb{E}(||B||_2)$?
The expected Frobenius norm of $B$ is defined to be
$$ \mathbb{E}(||B||_F)=\mathbb{E}\left(\sqrt{\sum_{i=1}^n\sum_{j=1}^n |b_{ij}|^2}\right) $$
where $b_{ij}$ are the elements of $B$.
The expected spectral norm of $B$ is defined to be
$$ \mathbb{E}(||B||_2)= \mathbb{E}\left(\max_{|x|_2 \ne 0}\frac{|Bx|_2}{|x|_2}\right). $$
Cross-posted to https://mathoverflow.net/questions/222994/how-to-calculate-expected-value-of-matrix-norms-of-ata now. I will update this question if anything substantive is added there.