Central Limit Theorem for Arm-Specific Means

Why the Allocation Fraction Enters the Limit

An arm-specific sample mean averages only the responses observed on that arm. Under adaptive assignment, the number of such responses is random and depends on the trial’s history. If the arm receives a stable, positive fraction of the assignments, its sample mean still has a Gaussian limit under suitable conditions. But its precision depends on that fraction: an arm receiving fewer observations has greater sampling variability.

The key is to separate two sources of randomness. The centered responses accumulated on an arm form a martingale sum, to which a martingale central limit theorem applies. The random number of observations then acts as a denominator. A positive limiting allocation fraction lets us replace that denominator, asymptotically, by a deterministic multiple of the total sample size.

Setup and Assumptions

Consider a trial with two or more arms, indexed by \(i\). At time \(t\), let \(A_t\) denote the assigned arm and \(Y_t\) the response observed after assignment. Let \(\mathcal{F}_{t-1}\) contain the information available just before that assignment. Adaptive assignment means that the distribution of \(A_t\) may depend on \(\mathcal{F}_{t-1}\); it does not mean that the assignment may use the current, not-yet-observed response.

For arm \(i\), suppose the response has conditional mean \(\mu_i\) and conditional variance \(\sigma_i^2\), given the past and assignment to arm \(i\). These quantities do not change with \(t\). The response noise must be sufficiently well behaved for a martingale central limit theorem: assume, for example, a uniformly bounded conditional \(2+\delta\) moment for some \(\delta>0\). This is a convenient sufficient condition for the conditional Lindeberg condition. More generally, the conditional Lindeberg condition itself can be assumed.

Define the number assigned to arm \(i\) by time \(n\) and its sample mean by

The sample mean is defined when \(N_i(n)>0\), which holds eventually under the positive-fraction condition below. Suppose that the allocation fraction converges in probability to a constant:

This condition says that the arm is sampled on the order of \(n\) times, not just that it is sampled infinitely often. The distinction matters for the rate and limit distribution.

Derivation by a Martingale Central Limit Theorem

Let \(\pi_{i,t}=P(A_t=i\mid\mathcal{F}_{t-1})\), and define the centered contribution at time \(t\) as \(D_{i,t}=I(A_t=i)(Y_t-\mu_i)\). Because assignment is made before observing the response, the conditional expectation of this contribution is zero:

Thus the cumulative centered response \(M_i(n)=\sum_{t=1}^{n}D_{i,t}\) is a martingale. Its predictable conditional variance is

The difference between the realized assignment count \(N_i(n)\) and \(\sum_{t=1}^{n}\pi_{i,t}\) is itself a martingale sum with increments \(I(A_t=i)-\pi_{i,t}\). These increments are bounded, so their sum divided by \(n\) converges to zero in probability. Consequently, if \(N_i(n)/n\) converges to \(\rho_i\), then \(\sum_{t=1}^{n}\pi_{i,t}/n\) does too. The conditional variance of \(M_i(n)/\sqrt{n}\) therefore converges to \(\rho_i\sigma_i^2\). With the conditional Lindeberg condition, the martingale central limit theorem gives

Since \(\bar{Y}_i(n)-\mu_i=M_i(n)/N_i(n)\), rescaling by \(\sqrt{n}\) yields

The final step uses \(n/N_i(n)\xrightarrow{P}1/\rho_i\) and Slutsky’s theorem. Equivalently, scaling by the arm’s own sample size gives the usual arm-level form:

Worked Examples

Assumptions, Diagnostics, and Alternatives

The central limit theorem is robust to adaptive assignment when the adaptation uses information available before the current response and the response-noise conditions hold. The following checks are useful when assessing whether its conclusion is credible:

• Check the sampling rate. The arm’s count should grow proportionally to \(n\), with a positive limiting fraction. A high allocation fraction at one interim analysis alone does not establish this asymptotic condition.
• Check response timing. Assignment based on past responses is permitted. Assignment that uses the current response, or otherwise selects participants based on unobserved current outcomes, can break the conditional mean-zero argument.
• Check the response model. A changing conditional mean or variance requires a different centering or variance calculation. Heavy tails that violate the Lindeberg condition can also prevent a Gaussian limit.
• Check variance estimation separately. The theorem states the limiting distribution when \(\sigma_i^2\) is the arm’s response variance. Using an estimated variance in a standard error requires that estimate to be consistent under the adaptive design.

The main alternative is the ordinary independent, identically distributed central limit theorem for a fixed sample from an arm. That argument treats the arm’s observations as a fixed-size sequence and does not explain how adaptive assignment and a random denominator enter the limit. The martingale argument handles the dependence of future assignments on past data. A conditional analysis given the realized allocation path can also be useful, but it must account for the response model and cannot simply assume that adaptively selected observations are an unconditional fixed random sample.

A positive limiting fraction is essential for the stated \(\sqrt{n}\) result. If the fraction tends to zero, then \(N_i(n)\) may still diverge, but the total-sample normalization and variance \(\sigma_i^2/\rho_i\) no longer apply. If the fraction does not settle to a constant, a single fixed normal variance need not describe the limit. These cases require separate analysis rather than substitution into this theorem.

Common Mistake