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: Welcome to another episode of ResearchPod.
Sam: Today we're looking at the paper "Almost sure null bankruptcy of testing-by-betting strategies" by Hongjian Wang, Shubhada Agrawal, and Aaditya Ramdas.
Alex: So what's the main puzzle here?
Sam: These betting strategies test if data matches a default assumption, called the null hypothesis—like checking if a website's click rate is exactly 10%. They work like a gambler betting pretend money on each click being above or below that 10%. The gambler's pile starts at 1 and grows or shrinks based on outcomes. If clicks truly average 10%, it's a fair game on average, but the paper shows the pile hits zero almost surely—meaning on nearly every possible run of data.
Alex: So the wealth tracks evidence against the null. It grows if clicks average something else, like 12%, but ruins if it's truly 10%?
Sam: Yes. Simple fixed bets fail in some cases, so researchers use adaptive ones like Krichevsky-Trofimov, or KT, and GRAPA. These adjust bet sizes based on past clicks, like checking the scoreboard to wager smarter next time.
Alex: They're learning from history. But they all bankrupt under the true null?
Sam: Exactly. The key is a sum-of-squares criterion. Each bet is a fraction of the current pile. If the sum of those fractions squared grows without bound over time, random ups and downs in the fair game multiply the pile down to zero almost surely. For KT, the fraction is the running total of past deviations from 10%, divided by the round number. Squared, it behaves like one over the round number most of the time, so the sum diverges.
Alex: Like the bets shrink slowly enough that squared noise piles up and drags wealth to zero.
Sam: Picture a drunkard's walk on a number line. After many steps, the position divided by the square root of steps hovers at some typical distance from zero, not hugging it. For KT, squared and scaled by one over steps, each term stays away from zero often enough for the sum to blow up.
Alex: And GRAPA does something similar?
Sam: Yes, GRAPA picks the fraction that would have grown the pile most in hindsight from past rounds. It also shrinks like one over square root of rounds. Hedging strategies bet half on above and half on below, shrinking even more like one over square root of rounds times log rounds—still enough for divergence.
Alex: So all these grow to infinity if the null is wrong, but hit zero if it's right.
Sam: Precisely. The paper covers predictable plug-in bets like KT and GRAPA, plus hedging. For mixtures, like universal portfolios, wealth converges to the weight on zero bet—which is zero if there's no cash held back.
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.