Copulas

Abstract

Abstract 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(x 1,x 2,...,x s) such that every k-dimensional face function g ( 1 , … , 1 , x i 1 , 1 , … , 1 , x i 2 , 1 , … , 1 , x i k , 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 i 1 x i 2 ...x i k 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.