Two-dimensional distribution function g(x, y) defined in [0, 1]2 is called copula, if g(x, 1) = x and g(1,y)= y for every x, y. Similarly, s-dimensional copula is a distribution function g(x1,x2,...,xs) such that every k-dimensional face function
g\left( {1, \ldots ,1,{x_{{i_1}}},1, \ldots ,1,{x_{{i_2}}},1, \ldots ,1,{x_{{i_k}}},1, \ldots ,1} \right)
is equal to xi1 xi2 ...xik for some but fixed k. In this paper we summarize and extend all known parts of copulas.
In this paper we use the following abbreviations:
{x} — fractional part of x;
{x} — x mod 1;
[x] — integer part of x;
u.d. — uniform distribution;
d.f. — distribution function;
a.d.f. — asymptotic distribution function;
u.d.p. — uniform distribution preserving;
step d.f. — step distribution function;
a.e. — almost everywhere;
#X — cardinality of the set X.