Establish asymptotic optimality of finite-horizon betting processes

Establish whether the finite-horizon counterparts of the GREE, GREL, and GREM betting processes are asymptotically optimal under general or practically relevant temporal scenarios, particularly when e-statistics or loss variables are not independent and identically distributed over time.

Background

The paper proves asymptotic optimality of the GREE, GREL, and GREM betting processes only under restrictive conditions, including independent and identically distributed observations or e-statistics and limited forms of forecast variation. These conditions are unlikely to hold in practical financial risk-management applications, especially for dependent ARIMA and GARCH-type time series.

Finite-horizon betting processes are introduced as potentially better suited to clustered periods of high risk and risk underestimation. The paper reports empirical evidence that they can improve detection in such settings, but it does not establish their asymptotic optimality. Determining whether they possess this property under broader dependence structures is therefore left unresolved.

References

Still, these topics would likely benefit from further investigation: For example, while we discussed in Theorem \ref{AsymOptiofAlgs} certain scenarios under which the GREE, GREL, or GREM betting processes are asymptotically optimal, these scenarios are highly theoretical and unlikely to apply well to practical applications. In particular, assumptions on the e-statistics or the loss variables being i.i.d. over time are highly unlikely to hold in practice. It thus seems reasonable to search for betting processes that are asymptotically optimal under more general and/or practically more relevant scenarios. One type of betting process we suggest considering for this is constituted by the finite-horizon counterparts of the aforementioned processes. While we could not make assertions about these processes being asymptotically optimal, they are likely to better detect underestimations for scenarios where high risks and underestimations of these risks occur clustered in time.

— On E-Backtesting: Generalizations and Sample Size Determination  (2609.05089 - Oestmann et al., 4 Sep 2026) in Section 8, Conclusion and Outlook

Moreover, the variance-sensitive 2/3 law is a first-order design result; exact finite-time optimal allocation for a specific betting boundary remains a numerical optimization problem.

— Target-Aware Sequential Inference: Pooled versus Stratified Anytime-Valid Designs  (2609.37873 - Hait, 29 Sep 2026) in Section 12.3, “Variance-adaptive tuning”