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

Abstract

Let $p,q\in (0,\infty ]$ and ${\ell }_p^m({\ell }_q^n)$ be the mixed-norm sequence space of real matrices $x=(x_{i,j}{)}_{i\le m,j\le n}$ 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 ${\mathbb{B}}_{p,q}^{m,n}$.

45 pages

References in corpus (2)

• A Central Limit Theorem for Convex Sets
• Spectral flatness and the volume of intersections of $p$-ellipsoids