Copulas

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,...,x_s) such that every k-dimensional face function g(1,…,1,xi1,1,…,1,xi2,1,…,1,xik,1,…,1) 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 x_i1 x_i2 ...x_ik 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.