On approximation of copulas

Tomasz Kulpa

doi:10.1155/s0161171299222594

Abstract

New sufficient conditions for strong approximation of copulas, generated by sequences of partitions of unity, are given. Results are applied to the checkerboard and Bernstein approximations.

Highlights

A copula is a distribution function of a doubly stochastic measure μ on the unit square [0, 1]2, i.e., C(x, y) = μ([0, x] × [0, y]) for x, y ∈ [0, 1]
It is surprising that any copula, even the copula which relates a pair of independent random variables, can be approximated arbitrarily closely in the uniform sense by copulas which correspond to the deterministic dependence between a pair of random variables
Since the set of copulas is isomorphic to the set of Markov operators on L∞[0, 1], a strong convergence of copulas is defined by the strong convergence of the corresponding Markov operators

Summary

Introduction

In [5, 6], Li, Mikusinski, Sherwood, and Taylor discussed sequences of approximation copulas given by partitions of unity. We say that T : L∞[0, 1] → L∞[0, 1] is a Markov operator if it satisfies the following three conditions Let kn : [0, 1]2 → R be a sequence of nonnegative measurable functions satisfying the following two conditions kn(x, y)dx = 1 kn(x, y)dy = 1 for a.e. y ∈ [0, 1]; for a.e. x ∈ [0, 1].

Objectives

Results

Conclusion