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On Iterated Lorenz Curves with Applications: The Multivariate Case

Vilimir Yordanov

Abstract

It is well known that a Lorenz curve, derived from the distribution function of a random variable, can itself be viewed as a probability distribution function of a new random variable [4]. In a previous work of ours [26], we proved the surprising result that a sequence of consecutive iterations of this map leads to a non-corner case convergence, independent of the initial random variable. Namely, the limiting distribution follows a power-law distribution. In this paper, we generalize our result to the multivariate setting. We do so using Arnold's type definition [4] of a Lorenz curve, which offers the greatest parsimony among its counterparts. The situation becomes more complex in higher dimensions as the map affects not only the marginals but also their dependence structure. Nevertheless, we prove the equally surprising result that under reasonable restrictions, the marginals again converge uniformly to a power-law distribution, with an exponent equal to the golden section. Furthermore, they become independent in the limit. To emphasize the multifaceted nature of the problem and broaden the scope of potential applications, our approach utilizes a variety of mathematical tools, extending beyond very specialized methods.

On Iterated Lorenz Curves with Applications: The Multivariate Case

Abstract

It is well known that a Lorenz curve, derived from the distribution function of a random variable, can itself be viewed as a probability distribution function of a new random variable [4]. In a previous work of ours [26], we proved the surprising result that a sequence of consecutive iterations of this map leads to a non-corner case convergence, independent of the initial random variable. Namely, the limiting distribution follows a power-law distribution. In this paper, we generalize our result to the multivariate setting. We do so using Arnold's type definition [4] of a Lorenz curve, which offers the greatest parsimony among its counterparts. The situation becomes more complex in higher dimensions as the map affects not only the marginals but also their dependence structure. Nevertheless, we prove the equally surprising result that under reasonable restrictions, the marginals again converge uniformly to a power-law distribution, with an exponent equal to the golden section. Furthermore, they become independent in the limit. To emphasize the multifaceted nature of the problem and broaden the scope of potential applications, our approach utilizes a variety of mathematical tools, extending beyond very specialized methods.

Paper Structure

This paper contains 45 sections, 73 theorems, 490 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Two-dimensional Lorenz curve iteration
  3. Main result
  4. Analysis of special cases
  5. Fréchet-Hoeffding upper-bound
  6. Fréchet-Hoeffding lower-bound
  7. Discussion
  8. Multivariate setting
  9. Setup and notation
  10. Main theorem: long and short formulations
  11. Extreme dependence
  12. Extreme positive dependence: Fréchet--Hoeffding upper-bound $M_{d}$
  13. The dependence concept and its invariance
  14. The $d$--dimensional marginal recursion
  15. Fixed point within the comonotone class
  16. ...and 30 more sections

Key Result

Theorem 3

Given the above definition of a bivariate Lorenz curve and its associated operator, if for $n=0,1,..$we consider the sequence $L_{n+1}^{F}(x_{1},x_{2}),$ defined by or equivalently by where in $L$ the subscript $n$ denotes the iteration step, while the superscript indicates starting distribution $F$, we have that $L_{n}^{F}(x_{1},x_{2})$ are by themselves distribution functions (with the possible

Figures (15)

  • Figure 1: Figure 1: $L_{1}^{n}(x_{1})$ evolution
  • Figure 2: Figure 2: $L_{2}^{n}(x_{2})$ evolution
  • Figure 3: Figure 3: Compounds of the inverse marginal d.f.s – $\Phi_{n}^{1}(x)$
  • Figure 4: Figure 4: Compounds of the inverse marginal d.f.s – $\Phi_{n}^{2}(x)$
  • Figure 5: Figure 5: Dependence measures
  • ...and 10 more figures

Theorems & Definitions (169)

  • Definition 1
  • Claim 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Conjecture 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Remark 10
  • ...and 159 more