Paper: arXiv 2609.31345

Authors: Florian Gach, Simon Hochgerner

Abstract

This article is concerned with the asymptotic shape of quantile surfaces, defined as the set of quantiles at a given level $α$ generated by a controlled one-dimensional distribution. Specifically, when the distribution arises as a linear combination of log-normal random variables and the control is a vector of positive coefficients, we prove that quantile surfaces are globally concave in the left tail ($α\to0$) and globally convex in the right tail ($α\to1$). Moreover, these surfaces exhibit asymptotic separation of scale and shape.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly mathematical and theoretical analysis of quantile surface properties, particularly for log-normal distributions. While the mathematical derivations are rigorous and novel, there is a distinct lack of empirical validation or backtesting, placing it firmly in the ‘Lab Rats’ quadrant. The findings have strong implications for risk management and portfolio optimization, but their practical application would require further empirical work.

Research Flowchart

  flowchart TD
    A[Research Goal: Asymptotic Shape of Quantile Surfaces] --> B{Methodology: Proof-based Analysis};
    B --> C[Inputs: Linear Combination of Log-normal RVs, Vector of Positive Coefficients];
    C --> D{Computational Process: Asymptotic Analysis, Limit Theory};
    D --> E[Outcome 1: Quantile Surfaces Globally Concave (α→0)];
    D --> F[Outcome 2: Quantile Surfaces Globally Convex (α→1)];
    D --> G[Outcome 3: Asymptotic Separation of Scale and Shape];