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An extremal problem in uniform distribution theory. (English) Zbl 06846993
Let \(x_n\) and \(y_n\), \(n= 1,2,\dots\) be uniformly distributed sequences in the unit interval and \(F\) be a given continuous function on \([0,1]^2\). A classical problem is the study of extremal limits of the form \(\frac1N\sum_{n=1}^NF(x_n, y_n)\), where \(N\to\infty\). It is equivalent to find optimal bounds for Riemann-Stieltjes integrals of the form \(\int_0^1\int_0^1 F(x, y)dC(x, y)\), where \(C\) is the asymptotic distribution function of the sequence \((x_n, y_n)\) and it is usually referred to as copula. As pointed out in [the reviewer and O. Strauch, Unif. Distrib. Theory 6, No. 1, 101–125 (2011; Zbl 1313.11089)] the solution of this problem depends on the sign of the partial derivative \(\frac{\partial^2F(x,y)}{\partial x\partial y}\). The main result of the paper is the following: Let \(0< x_1< x_2<1\) and \[ F(x, y) =\begin{cases} F_1(x, y) &\text{if }x\in (0, x_1),\frac{\partial^2F_1(x,y)}{\partial x\partial y}>0,\\ F_2(x, y) &\text{if }x\in (x_1, x_2),\frac{\partial^2F_2(x,y)}{\partial x\partial y}<0,\\ F_3(x, y) &\text{if }x\in (x_2,1),\frac{\partial^2F_3(x,y)}{\partial x\partial y}>0.\end{cases} \] Then the copula maximizing \(\int_0^1\int_0^1 F(x, y)d\tilde{C}(x, y)\) has the form \[ C(x, y) =\begin{cases} \min(x, h_1(y))&\text{if }x\in [0, x_1],\\ \max(x+h_2(y)-x_2, h_1(y)) &\text{if }x\in [x_1, x_2],\\ \min(x-x_2+h_2(y), y)&\text{if }x\in [x_2,1],\end{cases} \] where \(h_1(y) =C(x_1, y)\), \(h_2(y) =C(x_2, y)\) and \((h_1, h_2)\) satisfy the Euler-Lagrange differential equations. The authors also discuss connections of extremal limits of couples to the theory of optimal transport. In the final section, they solve the example \(F(x, y) = \sin(\pi(x+y))\) and relate this problem to combinatorial optimization based on the work of L. Uckelmann [in: Distributions with given marginals and moment problems. Proceedings of the 1996 conference, Prague, Czech Republic. Dordrecht: Kluwer Academic Publishers. 275–281 (1997; Zbl 0907.60022)].

MSC:
11K06 General theory of distribution modulo \(1\)
49Q20 Variational problems in a geometric measure-theoretic setting
60E05 Probability distributions: general theory
60A10 Probabilistic measure theory
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