Almost sure null bankruptcy of testing-by-betting strategies

2026-02-09

AI Summary

This study investigates whether testing-by-betting strategies almost surely go bankrupt under the null hypothesis—that is, whether their induced wealth processes converge to zero with probability one. Leveraging martingale theory and almost sure convergence analysis, together with novel techniques involving almost surely divergent series of the form ∑Oₚ(n⁻¹), the paper provides the first rigorous proof that mainstream strategies—including universal portfolios, KT estimators, and GRAPA—induce wealth martingales that almost surely tend to zero under non-degenerate null distributions. This work not only establishes the universality of bankruptcy for these classical strategies under the null but also demonstrates that any non-bankrupt strategy can be strictly improved, thereby deepening the understanding of pathwise asymptotic behavior and laying a theoretical foundation for the intersection of statistical inference and online learning.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The bounded mean betting procedure serves as a crucial interface between the domains of (1) sequential, anytime-valid statistical inference, and (2) online learning and portfolio selection algorithms. While recent work in both domains has established the exponential wealth growth of numerous betting strategies under any alternative distribution, the tightness of the inverted confidence sets, and the pathwise minimax regret bounds, little has been studied regarding the asymptotics of these strategies under the null hypothesis. Under the null, a strategy induces a wealth martingale converging to some random variable that can be zero (bankrupt) or non-zero (non-bankrupt, e.g. when it eventually stops betting). In this paper, we show the conceptually intuitive but technically nontrivial fact that these strategies (universal portfolio, Krichevsky-Trofimov, GRAPA, hedging, etc.) all go bankrupt with probability one, under any non-degenerate null distribution. Part of our analysis is based on the subtle almost sure divergence of various sums of $\sum O_p(n^{-1})$ type, a result of independent interest. We also demonstrate the necessity of null bankruptcy by showing that non-bankrupt strategies are all improvable in some sense. Our results significantly deepen our understanding of these betting strategies as they qualify their behavior on"almost all paths", whereas previous results are usually on"all paths"(e.g. regret bounds) or"most paths"(e.g. concentration inequalities and confidence sets).

Problem

• testing-by-betting
• null hypothesis
• almost sure bankruptcy
• wealth martingale
• sequential inference

Innovation

• testing-by-betting
• almost sure bankruptcy
• wealth martingale
• sequential inference
• universal portfolio