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} —1; x mod

[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