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

Abstract

Let $p,q\in (0,\mathrm{\infty }]$ and ${\ell }_p^m({\ell }_q^n)$ be the mixed-norm sequence space of real matrices $x={(x_{i,j})}_{i\le m,j\le n}$ endowed with the (quasi-)norm $\Vert x{\Vert }_{p,q}:={\Vert {(\Vert {(x_{i,j})}_{j\le n}{\Vert }_q)}_{i\le m}\Vert }_p$. We shall prove a Poincaré–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 ${\mathbb{B}}_{p,q}^{m,n}$.