Limit theorems for mixed-norm sequence spaces with applications to volume distribution

Let $p,q\in (0,\mathrm{\infty }]$ and ${\mathrm{\ell }}_p^m({\mathrm{\ell }}_q^n)$ be the mixed-norm sequence space of real matrices $x=(x_{i,j}{)}_{i\le m,j\le n}$ endowed with the (quasi-)norm $\mathrm{\parallel }x{\mathrm{\parallel }}_{p,q}:=\parallel (\mathrm{\parallel }(x_{i,j}{)}_{j\le n}{\mathrm{\parallel }}_q{)}_{i\le m}{\mathrm{\parallel }}_p$. We shall prove a Poincaré-Maxwell-Borel lemma for suitably scaled matrices chosen uniformly at random in the ${\mathrm{\ell }}_p^m({\mathrm{\ell }}_q^n)$ unit balls ${\mathbb{B}}_{p,q}^{m,n}$, and obtain both central and non-central limit theorems for their ${\mathrm{\ell }}_p({\mathrm{\ell }}_q)$-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 ${\mathbb{B}}_{p,q}^{m,n}$.