Almost sure null bankruptcy of testing-by-betting strategies

Paper: arXiv 2602.08888

Authors: Hongjian Wang, Shubhada Agrawal, Aaditya Ramdas

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 $\sum_n O_p(n^{-1})$ 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).

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper offers a rigorous theoretical analysis of sequential testing-by-betting algorithms, proving a fundamental almost sure bankruptcy result under the null hypothesis. While highly strong mathematically and conceptually novel, it is entirely theoretical with no empirical backtesting or market data experiments.

Research Flowchart

  flowchart TD
    A["Research Question: Do optimal testing-by-betting strategies go bankrupt under non-degenerate nulls?"] --> B["Inputs: Betting Strategies (Universal Portfolio, KT, GRAPA, Hedging) & Null Martingales"]
    B --> C["Asymptotics & Divergence Analysis: Evaluate sums of type ∑ O_p(1/n)"]
    C --> D["Improvability Analysis: Examine properties of non-bankrupt strategies"]
    D --> E["Finding 1: Almost Sure Null Bankruptcy (Strategies go bankrupt with probability 1 under null)"]
    D --> F["Finding 2: Necessity of Bankruptcy (Non-bankrupt strategies are strictly improvable)"]
    E --> G["Outcome: Characterized 'almost all paths' behavior, bridging sequential inference & online learning"]
    F --> G

Cited References (128)

1. Supplement to ``On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds’’ (2025)
2. Nonparametric iterated-logarithm extensions of the sequential generalized likelihood ratio test (2021)
3. Three test statistics for a nonparametric one-sided hypothesis on the mean of a nonnegative variable (2005)
4. Top two algorithms revisited (2022)
5. An Asymptotically Optimal Bandit Algorithm for Bounded Support Models. (2010)
6. Symmetric location estimation/testing by empirical likelihood (2004)
7. Adjusted empirical likelihood and its properties (2008)
8. Empirical Likelihood Ratio Confidence Regions (1990) — link

1 of 128 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.