Iterated Lorenz Curve Mapping

• Iterated Lorenz Curve Mapping is a process that repeatedly applies the Lorenz transformation to a distribution, yielding a unique fixed point characterized by power laws.
• The methodology involves recursive bounding and fixed-point analysis, where the distribution converges uniformly to x^φ, with φ approximating 1.618 (the golden section).
• This mapping has practical implications in measuring income inequality, constructing robust risk measures, and analyzing network degree distributions.

The iterated Lorenz curve mapping refers to the iterative application of a Lorenz-curve transformation to a distribution function, resulting in a sequence of new probability distributions. Originating from Arnold’s definition, this map exhibits remarkable convergence properties: under general conditions, repeated application leads to a stable, nontrivial fixed-point distribution that universally emerges, independent of the initial law. In both univariate and bivariate cases, the limiting behavior involves explicit power laws governed by the golden section, with deep implications for inequality measurement, risk modeling, and network theory.

1. Formal Definition and Operator Construction

Let $X$ be a non-negative random variable with distribution function $F(x)$ on $[0, \infty)$ and finite mean $\mu_F = \int_0^\infty (1 - F(u)) \, du < \infty$. The (primal) Lorenz curve is the function $L_F: [0,1] \rightarrow [0,1]$ given by

$$L_F(p) = \frac{1}{\mu_F} \int_0^p F^{-1}(u) \, du, \quad 0 \leq p \leq 1$$

where the generalized inverse $F^{-1}(u) = \inf\{x : F(x) \geq u\}$ for $u \in [0,1]$. Arnold's insight is that $L_F$ itself forms a distribution function on $[0,1]$, so the mapping

$L: F \mapsto L[F]$

defines an operator $$L: \text{CDFs on } [0, \infty) \rightarrow \text{CDFs on } [0,1]$$. By iteratively applying $L$, starting from $F_0 = F$, the sequence is defined recursively as

$F_{n+1} = L[F_n], \qquad n = 0,1,2,\ldots$

This recursive mapping—the iterated Lorenz curve mapping—produces a sequence of distribution functions with rich limiting structure (Ignatov et al., 24 Jan 2024, Yordanov, 2 Jul 2025).

2. Iterative Convergence and Fixed-Point Analysis

The main finding is that, for any initial non-negative $F_0$ with finite mean, the sequence $\{F_n\}$ converges uniformly on $[0,1]$ to a unique fixed point $G$ satisfying the self-referential equation

$G(x) = \frac{1}{\mu_G} \int_0^x G^{-1}(u) \, du$

This fixed point emerges as the solution to a nonlinear functional equation, which under further analysis produces explicit power-law forms.

For the "primal" Lorenz operator, the limit is

$$G(x) = x^\varphi, \quad \varphi = \frac{1+\sqrt{5}}{2} \approx 1.618$$

which is the golden section. The fidelity of the convergence is established via bounding sequences

$$\begin{cases} x^{a_n+1} \leq F_n(x) \leq x^{a_n}, & n \text{ odd} \ x^{a_n} \leq F_n(x) \leq x^{a_n+1}, & n \text{ even} \end{cases}$$

where $a_{n+1} = 1 + \frac{1}{a_n}$, converging to $\varphi$. This places $F_n(x)$ between two powers tending to $\varphi$, and thus $F_n \to x^\varphi$ uniformly (Ignatov et al., 24 Jan 2024).

3. Limiting Distributions and Parent Laws

The limiting distribution has density

$g(x) = \frac{d}{dx}G(x) = \varphi\,x^{\varphi-1}$

The parent distribution from which this Lorenz law arises—obtained by inverting the Lorenz curve mapping—is the Pareto type: $F(x) = 1 - x^{-\varphi}, \quad x \geq 1$ with density

$f(x) = \varphi\,x^{-\varphi-1}$

Thus, both the iterated-limit curve $G$ and its parent $F$ are governed by the golden-ratio exponent.

The "reflected" Lorenz-curve operator $L'$—constructed by a stationary-excess (equilibrium) transform followed by diagonal reflection: $$H_F(p) = \frac{1}{\mu_F}\int_p^1 F^{-1}(u)\,du;\qquad L'[F](x) = 1 - H_F(1-x)$$ converges uniformly to

$$G_{\text{ref}}(x) = 1 - (1-x)^{\varphi'}, \quad \varphi' = \frac{\sqrt{5}-1}{2} \approx 0.618$$

with density

$g_{\text{ref}}(x) = \varphi'\,(1-x)^{\varphi'-1}$

The parent law for the reflected case is also a Pareto form, with exponents $\varphi$ and $\varphi'$ satisfying $\varphi' = 1/\varphi$ and $1+\varphi = \varphi^2$.

4. Bivariate and Multivariate Extensions

Arnold’s bivariate Lorenz curve generalizes the construction for $X=(X_1,X_2) \geq 0$, with joint distribution $F_{12}$ and cross-moment $\mu_{12} = \mathbb{E}[X_1 X_2]$. The bivariate Lorenz curve is

$$L_{F_{12}}(x_1, x_2) = \frac{1}{\mu_{12}} \int_{u_1=0}^{F_1^{-1}(x_1)} \int_{u_2=0}^{F_2^{-1}(x_2)} u_1 u_2 \, dF_{12}(u_1,u_2)$$

Bivariate iteration involves nontrivial interdependence, as the Lorenz operator affects both marginals and their dependence structure (copula). The iteration is

$$L_{n+1}(x_1,x_2) = \frac{1}{D_n} \int_{u_1=0}^{L_{n,1}^{-1}(x_1)} \int_{u_2=0}^{L_{n,2}^{-1}(x_2)} u_1 u_2\,dL_n(u_1,u_2)$$

where $D_n$ is the normalization. For marginals,

$$L_{n+1,i}(x) = \frac{1}{D_n}\int_0^{L_{n,i}^{-1}(x)}u\,E[X_j^{L_n}| X_i^{L_n}=u]\,du$$

Under positive dependence (TP$_2$) or negative dependence (RR$_2$), the iteration preserves TP$_2$/RR$_2$ and induces uniform convergence of marginals to the power law governed by the golden section, with asymptotic independence in the joint density as the copula density approaches unity away from boundaries (Yordanov, 2 Jul 2025).

5. Differential Systems, Functional Equations, and Exponent Characterization

The limiting Lorenz and reflected curves solve nontrivial functional equations arising in self-similar differential systems. Specifically,

$$G'(x) = \frac{1}{\mu} (G^{-1}(x)), \qquad G(0)=0,\,G(1)=1$$

has unique solution for the primal fixed point, and

$$G_{\text{ref}}'(x) = \frac{1}{\mu}(1-G_{\text{ref}}^{-1}(1-x))$$

for the reflected. The solution form $G(x) = x^{1/\alpha}$ requires $\alpha^2 = \alpha + 1$; thus the exponent is $\varphi = \frac{1+\sqrt{5}}{2}$, the golden section.

A summary table for limiting cases:

Case	Limiting CDF	Exponent	Parent Law
Primal	$x^\varphi$	$\varphi$	Pareto, $-\varphi$
Reflected	$1-(1-x)^{\varphi'}$	$\varphi'$	Pareto, $1+\varphi$
Bivariate	$x^{1/\varphi}$	$1/\varphi$	Asymptotic indep.

6. Applications and Contexts of Use

Iterated Lorenz curve mapping provides a universal attractor for income-inequality modeling: convex combinations of the primal and reflected limits yield exceptional fits to real-world income data (e.g., Bulgarian household incomes). The limit power laws serve as benchmark solutions for self-similar differential equations relevant in economics and dynamical systems.

In risk management, bespoke risk measures—such as GS$_1$, GS$_2$—are built from the area between optimal (primal/reflected) Lorenz curves and a portfolio’s actual curve. These measures interpolate smoothly among mean-variance, CVaR, mean-absolute-deviation, and Gini-difference metrics, offering robust risk-return trade-offs and improved diversification profiles.

In network science, iterated Lorenz mapping offers mechanisms by which rich-get-richer transformations lead to degree-distribution tails with Pareto exponents $\varphi + 1$. Connections to stochastic-order transfers and queueing-theory stationary-excess transformations contextualize the mapping within broader mathematical frameworks (Ignatov et al., 24 Jan 2024).

A plausible implication is that the iterated mapping could underlie the universality of power laws in empirical distributions arising from repeated balancing or transfer operations.

7. Mathematical Foundations and Extensions

Proofs leverage Stieltjes integration, properties of TP$_2$/RR$_2$ densities, fixed-point functional analysis, and contraction mapping principles. Rearrangement inequalities (Hardy–Littlewood–Pólya) facilitate explicit bounds and uniform convergence.

In the bivariate case, the mapping’s effect on dependence structure is central: under iteration, even initially dependent marginals converge to independence, with copula density tending to 1 off-set boundaries. Extension to higher dimensions and multivariate settings via copula- or zonoid-based methods is ongoing. Full generalization to the multivariate case remains an open challenge (Yordanov, 2 Jul 2025).

Common misconceptions include the belief that Lorenz curve iteration always preserves dependence or scale parameters; the developed theory demonstrates the contrary: dependence structure erodes and universal limiting forms supersede the original law’s details.

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In summary, iterated Lorenz curve mapping defines a robust transformation on probability distributions, yielding universal limiting power-law forms characterized by the golden section, with extensive applications in socioeconomic modeling, risk theory, and network science. Its mathematical structure unifies approaches across distribution theory, functional analysis, and applied probability.