On Two-Dimensional Sequences Composed by One-Dimensional Uniformly Distributed Sequences

Let \(x_n\) and \(y_n\), \(n = 1, 2, \ldots\), 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 points of sequence
\[
\frac{1}{N} \sum_{n=1}^{N} F(x_n, y_n), \qquad N = 1, 2, \ldots.
\]
A basic idea is to apply distribution functions \(g(x,y)\) of the
two-dimensional sequence \((x_n, y_n)\), \(n = 1, 2, \ldots\). 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 \(\displaystyle\int_0^1 \int_0^1 F(x,y)\, d_x d_y\, g(x,y)\)
assuming that the differential \(d_x d_y F(x,y)\) has constant sign, and
also for \(F(x,y) = f(x)y\) where \(f(x)\) is a piecewise linear function.