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

Let p,q∈(0,∞] and ℓpm(ℓqn) be the mixed-norm sequence space of real matrices x=(xi,j)i≤m,j≤n endowed with the (quasi-)norm ‖x‖p,q:=‖(‖(xi,j)j≤n‖q)i≤m‖p. We shall prove a Poincaré–Maxwell–Borel lemma for suitably scaled matrices chosen uniformly at random in the ℓpm(ℓqn)-unit balls Bp,qm,n, and obtain both central and non-central limit theorems for their ℓp(ℓq)-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 Bp,qm,n.