Limit theorems for mixed-norm sequence spaces with applications to volume distribution (Q6595711)

Letting \(p,q\in(0,\infty]\) and \(m,n\in\mathbb{N}\), the authors consider the mixed-norm sequence spaces \(\ell_p^m(\ell_q^n)\) given by \(\mathbb{R}^{m\times n}\) endowed with the \(mn\)-dimensional Lebesgue measure and equipped with the quasinorm\N\[\N\lVert x\rVert_{p,q}=\lVert(\lVert(x_{i,j})_{j\leq n}\rVert_q)_{i\leq m}\rVert_p\,,\N\]\Nwhere \(\lVert\cdot\rVert_p\) is the usual \(\ell_p\)-norm. In particular, they establish a variety of central and non-central limit theorems for \(X=(X_{i,j})_{i\leq m,j\leq n}\), a random matrix uniformly distributed on the unit balls in these spaces, making use of a new probabilistic representation of this uniform distribution. These results include Poincaré-Maxwell-Borel principles giving limiting distributions for the first few coordinates of a random vector, and weak limit theorems. Three different regimes are considered: where \(m\to\infty\) while \(n\) is fixed, where \(n\to\infty\) while \(m\) is fixed, and where \(m\) depends on \(n\) and goes to infinity as \(n\to\infty\). An application is given to asymptotic volume distributions in the intersection of two mixed-norm sequence balls.