On two-dimensional sequences composed by one-dimensional uniformly distributed sequences (Q2893490)

Let \(x_n\) and \(y_n\), \(n=1,2,\dots \), be uniformly distributed sequences in the unit interval \([0,1)\) and let \(F(x,y)\) be a continuous function defined on \([0,1]^2\). The authors motivated by \textit{F. Pillichshammer} and \textit{S. Steinerberger} [Unif. Distrib. Theory 4, No. 1, 51--67 (2009; Zbl 1208.11088)] study extremes of limit points of the sequence \(\frac {1}{N}\sum _{n=1}^NF(x_n,y_n)\, N=1,2,\dots \). They transformed the problem by Helly theorems to compute extremes of NEWLINE\[NEWLINE\int _0^1\int _0^1 F(x,y)\,d_x\,d_y g(x,y)\tag{*}NEWLINE\]NEWLINE over all copulas \(g(x,y)\). The authors main results:NEWLINENEWLINENEWLINE1) If the partial derivatives \(d_xd_yF(x,y)>0\) for all \((x,y)\in [0,1]^2\), then by Fréchet-Hoeffding bounds the authors prove that the maximum of (*) over all copulas \(g(x,y)\) is \(\int _0^1F(x,x)\,dx\) and the minimum is \(\int _0^1F(x,1-x)\,dx\).NEWLINENEWLINE NEWLINE2) By the Sklar theorem they have similar extremes of (*) over all distribution functions \(g(x,y)\) with fixed margins.NEWLINENEWLINE NEWLINE3) The authors also give extremes of (*) if \(d_xd_yF(x,y)>0\) for all \((x,y)\in [0,1]\times [0,Y]\) and \(d_xd_yF(x,y)<0\) for all \((x,y)\in [0,1]\times [Y,1]\). Here \(Y\in (0,1)\) is arbitrary.NEWLINENEWLINE NEWLINE4) Finally, they find extremes of the integral \(\int _0^1f(x)\Phi (x)\,dx\) over all uniform distribution preserving function \(\Phi (x)\), which was also considered by \textit{S. Steinerberger} [Unif. Distrib. Theory 4, No. 1, 117--145 (2009; Zbl 1208.11089)].