This post is a purely mathematical thought experiment about stochastic processes and stopping times. It is not gambling advice, nor an endorsement of gambling. The setup uses gambling language because it provides an intuitive narrative for a probabilistic phenomenon. 

Consider a sequence of repeated gambling decisions, each producing either a gain of one unit or a loss of one unit. Denote these outcomes by random variables  

And the cumulative outcome

Thus Xn records the total net gain (or loss) after n rounds of gambling. When the probability of a gain equals the probability of a loss (p = 0.5), the process is “fair” in the probabilistic sense and Xn forms a martingale. If gains are more likely than losses (p > 0.5), the process becomes a submartingale; if losses are more likely, it becomes a supermartingale.  

In any sequential setting, the rule for when to stop is as important as the dynamics of the increments. Even if the per-round outcomes ξi​ are simple, the cumulative process Xn​ can behave very differently depending on whether one runs it indefinitely or terminates it according to a data-dependent rule. This is captured by the stopping time T. Formally, a random time T is called a stopping time if the event {T=n} depends only on information available up to time n. That is, the decision to stop at time n can be made without knowing the future. Stopping rules are central because they determine not only whether we eventually stop, but also how long we expect to run:

In sequential systems, finiteness of  E[T]  is often more important than almost sure stopping. Finiteness of E[T] is a proxy for the feasibility of a strategy: it distinguishes processes that reach a goal in a controlled, predictable time scale from those that may reach it eventually but only after a heavy-tailed waiting time, where the expectation is infinite. This distinction is the lens through which the remainder of this post is written.

The Classical Martingale Case

Suppose having fair increments (p=0.5) which makes (Xn )n≥0 a martingale with respect to its natural filtration. A canonical stopping rule for this process would be the two–sided stopping rule:

meaning stop when cumulative gains reach a, or cumulative losses reach b.  In this setup, standard computation for uses the fact that the quadratic process:

is also a martingale, thus yielding:

Further, since (Xn∧T)n≥0 is a bounded martingale, E[XT]=0, and thus

Accordingly, in the fair martingale case, a two-sided stopping rule produces a process that terminates on a controlled time scale, with the expected duration growing linearly in each barrier. Removing any of the (upper or lower) barriers and adopting a one-sided stopping rule will result in E[T]=∞, namely, any guarantee of finishing in finite expected time disappears.

From Martingales to (Fading) Submartingales

The martingale model above is intentionally pessimistic: it represents a setting with no persistent advantage, where gains and losses balance in expectation. In many sequential decision problems, however, one may make decisions adaptively using information and strategy, thus making a small edge in the gain/loss-making probabilities. In such cases it is natural to model the cumulative process as a submartingale, with slightly positive expected increments. At the same time, it is rarely realistic to assume that an edge remains constant indefinitely. Performance can degrade with fatigue; opponents may adapt; the environment may drift; and any fixed strategy tends to be partially arbitraged away as it becomes predictable. These considerations motivate a model in which the advantage is present initially but decays over time, gradually reverting toward a fair random walk.

Formally, assume the increments ξn​ satisfy:

with f(n) ↓ 0 as n → ∞, representing the edge decadence. The natural temptation, once the process is a submartingale, is to abandon the lower loss barrier and adopt a one-sided stopping rule of the form: 

(meaning the gambler does not stop until they hit a certain profit a). The problem we study hereafter is for which rates of decadence of f(n) does E[T]<∞ hold, and for which rates does it fail.

The structure of pn above grants the following cumulative edge to the process:

Accordingly, the centered process of the form:

is a martingale satisfying finite increments:

Hence, by Azuma's inequality :

Now, observe that the event {T>n} implies, in particular, that Xn ≤ a−1, hence:

This inequality already suggests the governing principle: E[T] is finite if Σf(i) grows faster than a threshold. To expose the threshold cleanly, consider the polynomial model:

for large enough n, the model yields: