On two-dimensional sequences composed by one-dimensional uniformly distributed sequences

Let $x_n$ and $y_n$, $n=1,2,\dots$, be sequences in the unit interval $[0,1)$ and let $F(x,y)$ be a continuous function defined on $[0,1]^2$. In this paper we consider limit pointsof sequence $frac{1}{N}\sum_{n=1}^NF(x_n,y_n)$, $N=1,2,\dots$ A basic idea is to apply distribution functions $g(x,y)$of two-dimensional sequence $(x_n,y_n)$, $n=1,2,\dots$. It can be shown that every limit point has the form $\int_0^1 \int_0^1 F(x,y)d_x d_y g(x,y)$. If, moreover, both sequences $x_n$ and $y_n$ are uniformly distributed, then distribution functions $g(x,y)$ are called copulas and we find extremes of $\int_0^1 \int_0^1 F(x,y)d_x d_y g(x,y)$ assuming that the differential $d_x d_y F(x,y)$ has constant signum and also for $F(x,y)=f(x)y$ where $f(x)$ is a piecewise linear function.