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Asymptotic Classification Error for Heavy-Tailed Renewal Processes

Xinhui Rong, Victor Solo

TL;DR

This work derives the first asymptotic Bhattacharyya bound for binary classification of heavy-tailed renewal processes, where inter-event times lack moment-generating functions and survivors are regularly varying. Using Laplace-transform analysis and Karamata-Tauberian theory, the authors show that the Bhattacharyya bound decays as a near power law $B_*(T)\sim \frac{1}{1-c_{12}}T^{-\rho}L(T)$ with $\rho=(\rho_1+\rho_2)/2$, making heavy-tailed renewals harder to classify than regular renewals. The results are specialized to Pareto inter-event times, yielding a closed-form bound $B_*(T)=\frac{1}{(T/\alpha_2)^\beta}\frac{\theta^{\beta_1}}{1-\delta\theta^{\beta_1}}$, and validated via analytical and numerical simulations demonstrating convergence and practical relevance for determining observation length. The findings provide a rigorous, interpretable guide for expected error bounds and classification time in systems exhibiting heavy-tailed event timings, with implications for neural, social, and stochastic-process data.

Abstract

Despite the widespread occurrence of classification problems and the increasing collection of point process data across many disciplines, study of error probability for point process classification only emerged very recently. Here, we consider classification of renewal processes. We obtain asymptotic expressions for the Bhattacharyya bound on misclassification error probabilities for heavy-tailed renewal processes.

Asymptotic Classification Error for Heavy-Tailed Renewal Processes

TL;DR

This work derives the first asymptotic Bhattacharyya bound for binary classification of heavy-tailed renewal processes, where inter-event times lack moment-generating functions and survivors are regularly varying. Using Laplace-transform analysis and Karamata-Tauberian theory, the authors show that the Bhattacharyya bound decays as a near power law with , making heavy-tailed renewals harder to classify than regular renewals. The results are specialized to Pareto inter-event times, yielding a closed-form bound , and validated via analytical and numerical simulations demonstrating convergence and practical relevance for determining observation length. The findings provide a rigorous, interpretable guide for expected error bounds and classification time in systems exhibiting heavy-tailed event timings, with implications for neural, social, and stochastic-process data.

Abstract

Despite the widespread occurrence of classification problems and the increasing collection of point process data across many disciplines, study of error probability for point process classification only emerged very recently. Here, we consider classification of renewal processes. We obtain asymptotic expressions for the Bhattacharyya bound on misclassification error probabilities for heavy-tailed renewal processes.
Paper Structure (12 sections, 3 theorems, 13 equations, 2 figures)

This paper contains 12 sections, 3 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classification of Renewal Processes
  3. Renewal Processes
  4. Likelihood Classification and the Misclassification Error
  5. The Point Process Bhattacharyya Bound
  6. Asymptotic Bhattacharyya Bound
  7. Analytical and Numerical Studies
  8. Pareto Example
  9. Pareto Simulations
  10. Comparison with the bound
  11. Comparison with the error probability
  12. Conclusions

Key Result

Theorem 1

xr21 For renewal process es, $B(T)$ has LT where $\bar{p}_{12}(s)$ is the LT of $p_{12}(x) = \sqrt{p_1(x)p_2(x)}$ and $\bar{G}_{12}(s)$ is the LT of $G_{12}(x) = \sqrt{S_1(x)S_2(x)}$, and $S_k(x)$ is the survivor function of class $k$ IETs.

Figures (2)

  • Figure 1: Heatmap of $y=\ln[\hat{B}(T)/B_*(T)]$: We convert $15\times3\times5$ sets of results in one heatmap. We display the values of $y$ for the largest $T$ for each pair of $(\gamma,\theta)$. It is verified that $y\to0\Rightarrow B(T)\sim B_*(T)$.
  • Figure 2: Heatmap of $z=\frac{\ln[2\hat{P}_e(T)/B_*(T)]}{\beta\ln(T)}$: We demonstrate that the classification error probability $P_e(T)$ also has power decay, by showing that $z$ converges to a negative constant.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3