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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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