Almost sure null bankruptcy of testing-by-betting strategies

TL;DR

This paper proves that various testing-by-betting strategies almost surely go bankrupt under the null hypothesis, deepening understanding of their asymptotic behavior in statistical inference and online learning.

Contribution

It establishes that these strategies almost surely bankrupt under the null, providing new insights into their asymptotic properties and necessity of null bankruptcy.

Findings

All tested strategies go bankrupt with probability one under the null hypothesis.

The analysis reveals the almost sure divergence of certain sums of order 1/n, of independent interest.

Non-bankrupt strategies are shown to be improvable, highlighting the necessity of bankruptcy.

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).