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

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="normal">Λ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> ${\left\{{{\Lambda}}_{n}=\left\{{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right\}\right\}}_{n}$ be a sequence of finite multisets of real numbers such that d n → ∞ as n → ∞, and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msup> <m:mo>→</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:math> $f:{\Omega}\subset {\mathbb{R}}^{d}\to \mathbb{R}$ be a Lebesgue measurable function defined on a domain Ω with 0 &lt; μ d (Ω) &lt; ∞, where μ d is the Lebesgue measure in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{d}$ . We say that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="normal">Λ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> ${\left\{{{\Lambda}}_{n}\right\}}_{n}$ has an asymptotic distribution described by f , and we write <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="normal">Λ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>∼</m:mo> <m:mi>f</m:mi> </m:math> ${\left\{{{\Lambda}}_{n}\right\}}_{n}\sim f$ , if (*) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">