Hongjian Wang, Shubhada Agrawal, Aaditya Ramdas
4 min
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).
Alex: Like spreading bets across many guesses for the deviation, but only surviving if some money sits untouched?
Sam: Yes. Adding cash makes it less powerful under deviations, since you could wager that safe part too for faster growth. All good mixtures are cash-free, so they bankrupt.
Alex: A built-in trade-off: more power means certain ruin when the null holds.
Sam: The paper also shows that on paths where future wealth stays above some level based on past data, you can always improve power by removing safety margins—like borrowing against the safe part to bet more.
Alex: But that's only on predictable safe paths?
Sam: Yes, it leaves a gap for unpredictable ones, calling it preliminary insight. The logic extends to sub-Gaussian data, like bounded outcomes, using similar rules.
Alex: So bankruptcy is tied to strong power across these main strategies.
Sam: Overall, the paper reveals the inevitable cost of universal testing power: certain null bankruptcy under the true assumption. That's a meaningful symmetry for designing these tests.
Alex: Thanks for breaking it down so clearly, Sam. Listeners, that's our look at null bankruptcy in testing-by-betting strategies. Thanks for listening to ResearchPod.