Paper: arXiv 2610.41990

Authors: Yinhao Wu

Abstract

Does the deepest drawdown take the longest to recover? For random walks with iid increments of positive finite mean, we find that the deepest and longest episodes are eventually distinct almost surely under light-tailed losses, but coincide with probability tending to one under heavy-tailed losses. More precisely, when losses have a finite exponential moment, the deepest episode’s duration, relative to the longest among $n$ episodes, converges in probability to an explicit $ρ\in(0,1)$, while its descending duration rank is $n^{1-ρ+o_{\mathbb{P}}(1)}$. These limits hold for every fixed number of leading depths; in the Gaussian model, $ρ=1/2$, independently of drift and variance. The separation reflects the different exponential costs of reaching a large depth and sustaining a long episode. For regularly varying losses, a common large loss governs both depth and duration: the duration ratio tends to one almost surely, and for every fixed $k$, the $k$ deepest and $k$ longest episodes have the same indices in the same order with probability tending to one. For any $k_n\to\infty$ with $k_n=o(n)$, the proportion of common episodes among the $k_n$ deepest and $k_n$ longest converges in probability to zero in the light-tailed case and to one in the heavy-tailed case. These ranking laws also hold for completed drawdowns within a deterministic observation horizon.

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 is highly theoretical, focusing on rigorous mathematical proofs for random walk properties under different loss tail assumptions. While the mathematical derivations are complex and novel, there is no empirical validation or backtesting presented. The clarity is good for a mathematically inclined audience, but reproducibility is limited by the absence of code or data.

Research Flowchart

  flowchart TD
    A[Research Goal: When is the deepest drawdown also the longest?] --> B{Methodology: Analyze Random Walks with i.i.d. Increments};
    B --> C[Inputs: Loss Distributions (Light-Tailed vs. Heavy-Tailed)];
    C --> D{Computational Process: Asymptotic Analysis of Drawdown Depth & Duration};
    D --> E1[Outcome: Light-Tailed Losses -> Deepest and Longest Episodes are Distinct];
    D --> E2[Outcome: Heavy-Tailed Losses -> Deepest and Longest Episodes Coincide];
    E1 --> F1[Finding: Duration of Deepest relative to Longest -> Explicit $\rho \in (0,1)$];
    E2 --> F2[Finding: Duration Ratio Tends to One; Same Indices for k Deepest/Longest];