  Let $p, q \in (0, \infty]$ and $\ell_p^m(\ell_q^n)$ be the mixed-norm
sequence space of real matrices $x = (x_{i, j})_{i \leq m, j \leq n}$ endowed
with the (quasi-)norm $\Vert x \Vert_{p, q} := \big\Vert \big( \Vert (x_{i,
j})_{j \leq n} \Vert_q \big)_{i \leq m} \Vert_p$. We shall prove a
Poincar\'e-Maxwell-Borel lemma for suitably scaled matrices chosen uniformly at
random in the $\ell_p^m(\ell_q^n)$ unit balls $\mathbb{B}_{p, q}^{m, n}$, and
obtain both central and non-central limit theorems for their
$\ell_p(\ell_q)$-norms. We use those limit theorems to study the asymptotic
volume distribution in the intersection of two mixed-norm sequence balls. Our
approach is based on a new probabilistic representation of the uniform
distribution on $\mathbb{B}_{p, q}^{m, n}$.
