Rare-Arm Asymptotics With Shrinking Allocation Probabilities

Why a Shrinking Allocation Changes the Rate

In an adaptive trial, an arm can receive a smaller share of assignments as evidence accumulates. If its assignment probability decreases over time, the number of observations in that arm grows more slowly than the total trial size. That slower growth changes the precision of its response estimate: the usual square-root-of-total-sample-size scale no longer describes the arm-specific error.

This tutorial studies a polynomial schedule for a particular arm \(a\). The central calculation is from assignment probability to cumulative arm count, then from that count to the estimator’s limiting distribution. A Gaussian limit requires more than a diverging expected count: response moments and the contribution of individual observations must also be controlled.

Assumptions and the Arm-Specific Rate

For the main result, suppose \(0<\gamma<1\) and \(q_{i,a}/(c_a i^{-\gamma})\to 1\) almost surely, where \(c_a\) is nonrandom. The probabilities are predictable: each \(q_{i,a}\) is determined by information available before assignment \(i\). Assignments are made according to these probabilities. Assume the response observed from arm \(a\), when assigned, has mean \(\mu_a\), variance \(\sigma_a^2\), and a common response distribution across participants. Conditional on the history and assignment, its response follows that arm-specific distribution; the randomization does not change the response distribution within the arm.

For the Gaussian limit below, also assume that the arm-\(a\) responses have a finite \(2+\delta\) absolute central moment for some \(\delta>0\). More generally, a direct martingale Lindeberg condition can replace this moment assumption. Requiring a common distribution, or an appropriate uniform moment bound, matters: merely requiring each response to have a finite \(2+\delta\) moment does not ensure that extreme responses become negligible uniformly as the trial grows.

The cumulative assignment probability has the same leading order as the sum of \(c_a i^{-\gamma}\). For \(0<\gamma<1\), comparison with the corresponding integral gives \(\sum_{i=1}^n i^{-\gamma}\sim n^{1-\gamma}/(1-\gamma)\). The realized count \(M_a(n)\) concentrates around this diverging cumulative probability on its relative scale. Thus, with \(\kappa_a=c_a/(1-\gamma)\),

Here \(u_n\sim v_n\) means \(u_n/v_n\to1\). The arm’s sample size diverges, but it is of smaller order than \(n\): \(M_a(n)/n\to0\). Consequently, the arm-specific mean estimator \(\bar Y_{a,n}=M_a(n)^{-1}\sum_{i=1}^n I_{i,a}Y_{i,a}\), defined when \(M_a(n)>0\), has a slower rate than an estimator based on a fixed positive allocation fraction.

Deriving the Gaussian Limit

Write \(Y_{i,a}\) for the response that participant \(i\) would provide under arm \(a\), and consider the centered, observed contribution \(I_{i,a}(Y_{i,a}-\mu_a)\). Given the past, its conditional mean is zero and its conditional variance is \(q_{i,a}\sigma_a^2\). The sum of these contributions is a martingale sum: each new centered contribution has conditional mean zero given the information before that assignment.

The accumulated conditional variance is \(\sigma_a^2\sum_{i=1}^n q_{i,a}\), which is asymptotic to \(\sigma_a^2\kappa_a n^{1-\gamma}\). Under the stated moment condition, the martingale Lindeberg condition holds: the total contribution of increments larger than any fixed fraction of the normalization vanishes. In particular, the uniform \(2+\delta\) moment bound implied by the common response distribution controls the large increments as the normalization diverges. The martingale central limit theorem therefore gives a Gaussian limit for the centered numerator on the scale \(\sqrt{\kappa_a n^{1-\gamma}}\). Dividing by the arm count and using its rate yields the estimator limit:

The first expression uses the effective arm count scale; because \(M_a(n)/(\kappa_a n^{1-\gamma})\to1\), it is asymptotically equivalent to standardizing by \(\sqrt{M_a(n)}\). The second expression uses trial size \(n\). In practical terms, the standard error is approximately \(\sigma_a/\sqrt{\kappa_a n^{1-\gamma}}\), not \(\sigma_a/\sqrt{n}\).

Different Arms Can Have Different Rates

The polynomial exponent is arm-specific if the allocation schedules differ by arm. A larger exponent means that an arm’s probability shrinks faster and its sample size grows more slowly. The consequence is not just a different allocation proportion: the arm-specific estimator has a different convergence rate. The following hypothetical schedule illustrates the scale difference.

Conditions, Diagnostics, and Alternatives

The result relies on a predictable assignment mechanism whose probabilities follow the stated polynomial rate, a diverging arm count, stable arm-specific response means and variances, and a valid central limit argument for the response contributions. In an adaptive trial, the assignment probabilities can depend on earlier outcomes; the schedule need not be independent of the observed history. What matters here is that the probabilities are known given the past, have the specified asymptotic behavior, and the response assumptions and moment condition support the martingale limit.

Useful diagnostics include plotting the realized arm count against \(n^{1-\gamma}\), checking whether the recorded assignment probabilities follow the proposed schedule, and examining whether a few observed responses dominate the arm’s centered sum. The count plot is a diagnostic for a rate, not proof of the response central limit theorem. A very small realized count can make a normal approximation unreliable at the available trial size even when the asymptotic assumptions hold. Marked response skewness, extreme observations, or evidence that response variability changes over time should prompt scrutiny of the moment and stability assumptions.

A common mistake is to use the usual \(\sqrt n\) normalization because the trial has \(n\) participants. For this arm, the effective sample size is \(M_a(n)\) of order \(n^{1-\gamma}\), so the estimator’s error is of order \(n^{-(1-\gamma)/2}\). Another mistake is to infer a Gaussian limit just because each response separately has a finite higher moment. A uniform moment condition, a common response distribution as assumed here, or direct verification of the martingale Lindeberg condition is needed to rule out dominant increments.

With a fixed positive assignment probability, the arm count is of order \(n\) and the mean typically has the familiar \(\sqrt n\) rate. Polynomial shrinking preserves a diverging count when \(0<\gamma<1\), but at a slower rate. This analysis is specific to that range: it does not establish the same Gaussian limit for schedules with a different boundary behavior. Its practical limitation is that asymptotic rates describe large trials; they do not guarantee accurate finite-sample normal approximations when an arm has only a modest number of responses.