From asymptotic distribution and vague convergence to uniform convergence, with numerical applications

Let { Λ n = { λ 1 , n , … , λ d n , n } } n ${\left\{{{\Lambda}}_{n}=\left\{{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right\}\right\}}_{n}$ be a sequence of finite multisets of real numbers such that d _n → ∞ as n → ∞, and let f : Ω ⊂ R d → R $f:{\Omega}\subset {\mathbb{R}}^{d}\to \mathbb{R}$ be a Lebesgue measurable function defined on a domain Ω with 0 < μ _d (Ω) < ∞, where μ _d is the Lebesgue measure in R d ${\mathbb{R}}^{d}$ . We say that { Λ n } n ${\left\{{{\Lambda}}_{n}\right\}}_{n}$ has an asymptotic distribution described by f , and we write { Λ n } n ∼ f ${\left\{{{\Lambda}}_{n}\right\}}_{n}\sim f$ , if (*) lim n → ∞ 1 d n ∑ i = 1 d n F ( λ i , n ) = 1 μ d ( Ω ) ∫ Ω F ( f ( x ) ) d x ${\mathrm{lim}}_{n\to \infty }\frac{1}{{d}_{n}}{\sum }_{i=1}^{{d}_{n}}F\left({\lambda }_{i,n}\right)=\frac{1}{{\mu }_{d}\left({\Omega}\right)}{\int }_{{\Omega}}F\left(f\left(\boldsymbol{x}\right)\right)\mathrm{d}\boldsymbol{x}$ for every continuous function F with bounded support. If Λ _n is the spectrum of a matrix A _n , we say that { A n } n ${\left\{{A}_{n}\right\}}_{n}$ has an asymptotic spectral distribution described by f and we write { A n } n ∼ λ f ${\left\{{A}_{n}\right\}}_{n}{ \sim }_{\lambda } f$ . In the case where d = 1, Ω is a bounded interval, Λ _n ⊆ f (Ω) for all n , and f satisfies suitable conditions, Bogoya, Böttcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to 0 of the difference between the properly sorted vector [ λ 1 , n , … , λ d n , n ] $\left[{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right]$ and the vector of samples [ f ( x 1 , n ) , … , f ( x d n , n ) ] $\left[f\left({x}_{1,n}\right),\dots ,f\left({x}_{{d}_{n},n}\right)\right]$ , i.e., (**) lim n → ∞ max i = 1 , … , d n | f ( x i , n ) − λ τ n ( i ) , n | = 0 , ${\mathrm{lim}}_{n\to \infty } {\mathrm{max}}_{i=1,\dots ,{d}_{n}}\vert f\left({x}_{i,n}\right)-{\lambda }_{{\tau }_{n}\left(i\right),n}\vert =0,$ where x 1 , n , … , x d n , n ${x}_{1,n},\dots ,{x}_{{d}_{n},n}$ is a uniform grid in Ω and τ _n is the sorting permutation. We extend this result to the case where d ⩾ 1 and Ω is a Peano–Jordan measurable set (i.e., a bounded set with μ _d (∂Ω) = 0). We also formulate and prove a uniform convergence result analogous to (**) in the more general case where the function f takes values in the space of k × k matrices. Our derivations are based on the concept of monotone rearrangement (quantile function) as well as on matrix analysis arguments stemming from the theory of generalized locally Toeplitz sequences and the observation that any finite multiset of numbers Λ n = { λ 1 , n , … , λ d n , n } ${{\Lambda}}_{n}=\left\{{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right\}$ can always be interpreted as the spectrum of a matrix A n = diag ( λ 1 , n , … , λ d n , n ) ${A}_{n}=\mathrm{diag}\left({\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right)$ . The theoretical results are illustrated through numerical experiments, and a reinterpretation of them in terms of vague convergence of probability measures is hinted.