Joint Limit Laws for Allocation Counts and Response Estimates

Why a Joint Limit Is Needed

In an adaptive trial, the allocation rule can use accumulated responses to choose later assignments. Consequently, the final allocation counts and the estimated arm means are based on a shared evolving data history. Knowing a limit law for each quantity separately does not generally determine the limit law of the pair: their covariance can affect standard errors for combined quantities, such as a treatment contrast together with an allocation deviation.

The goal here is to express that dependence explicitly. We consider a fixed number \(K\) of treatment arms and a total of \(n\) assignments. Allocation counts are treated as a vector with one redundant coordinate removed, because the counts sum to \(n\). The response estimates are then obtained from the response-score components by accounting for their random denominators.

Setup and Assumptions

Write \(N_{1:K-1}(n)\) for the first \(K-1\) counts and \(\pi_{1:K-1}\) for the corresponding target fractions. The final count is determined by \(N_K(n)=n-\sum_{i=1}^{K-1}N_i(n)\); it is not an additional unconstrained coordinate. Let \(\mathcal{F}_{t-1}\) denote the observed trial history before assignment \(t\). The assignment probabilities are assumed predictable: \(q_{i,t}=P(A_t=i\mid\mathcal{F}_{t-1})\) is determined from that history before \(Y_t\) is observed.

For each arm, suppose the response innovation \(\varepsilon_{i,t}=Y_t-\mu_i\), when arm \(i\) is assigned, has conditional mean zero given the past and that assignment. A convenient sufficient variance condition is a finite, stable conditional second moment; more general conditional variances are allowed if their normalized sums converge. These assumptions describe the response noise, not independence of the assignments across time.

The central allocation condition is a joint asymptotic linear representation. Assume there are \((K-1)\)-dimensional martingale differences \(\psi_t\) such that \(N_{1:K-1}(n)-n\pi_{1:K-1}=\sum_{t=1}^n\psi_t+o_P(\sqrt n)\). This condition says that, at the square-root-of-\(n\) scale, the allocation-count fluctuation has a martingale representation. It is a substantive condition on the allocation procedure; convergence of \(N_i(n)/n\) to \(\pi_i\) alone does not imply it.

Define the response-score increment \(\xi_t=(I_{1,t}\varepsilon_{1,t},\ldots,I_{K,t}\varepsilon_{K,t})^\mathsf{T}\). Assume \((\psi_t^\mathsf{T},\xi_t^\mathsf{T})^\mathsf{T}\) is a martingale-difference vector satisfying a multivariate martingale central limit theorem: its conditional covariance sums divided by \(n\) converge in probability to a finite, nonrandom matrix, and its conditional Lindeberg condition holds. These conditions rule out a non-negligible contribution from rare, exceptionally large increments. Denote the limiting covariance blocks by \(\Omega_A\) for \(\psi_t\), \(C\) for the cross-covariance of \(\psi_t\) and \(\xi_t\), and \(V\) for \(\xi_t\).

The Joint Limit and Its Covariance

The dimensions clarify the result: \(\Omega_A\) is \((K-1)\) by \((K-1)\), \(C\) is \((K-1)\) by \(K\), and \(V\) is \(K\) by \(K\). The matrix \(D\) rescales each score by its arm's target allocation fraction. In particular, the cross-covariance between the count fluctuation and the estimated means is \(CD\), not necessarily zero.

Why the Denominators Produce This Form

The response score for arm \(i\) is \(S_i(n)=\sum_{t=1}^n I_{i,t}(Y_t-\mu_i)\). Whenever \(N_i(n)>0\), the estimator satisfies the exact identity \(\hat{\mu}_i(n)-\mu_i=S_i(n)/N_i(n)\). Thus, with \(S(n)=(S_1(n),\ldots,S_K(n))^\mathsf{T}\), \[ \sqrt n(\hat{\mu}(n)-\mu) =\operatorname{diag}\left(\frac{n}{N_1(n)},\ldots,\frac{n}{N_K(n)}\right)\frac{S(n)}{\sqrt n}. \] The allocation fractions converge to positive \(\pi_i\), so the diagonal matrix converges in probability to \(D\). The joint martingale central limit theorem applied to the allocation representation and \(S(n)=\sum_t\xi_t\) gives a Gaussian limit with covariance blocks \(\Omega_A,C,V\). Applying the continuous mapping theorem, or equivalently Slutsky's theorem to the convergent diagonal factor, transforms the score block into \(D\) times that block. The resulting covariance blocks are \(CD\) and \(DVD\), as claimed.

This also explains why analyzing the margins separately loses information. The score limit determines the response-estimate margin after rescaling, but it does not determine how that margin co-varies with the allocation limit. That dependence is carried by \(C\).

Contrasts and the Redundant Count

A joint limit is especially useful for linear summaries. If \(a\) is a vector of coefficients for a response contrast, then the scaled contrast error is \(a^\mathsf{T}\sqrt n(\hat{\mu}(n)-\mu)\). Its limiting variance is \(a^\mathsf{T}DVDa\), and its covariance with the count fluctuation is \(\Omega_A\) and \(CD\) combined through the corresponding linear transformations. For two arms, the second-arm count fluctuation is exactly the negative of the first-arm fluctuation, since \[ \frac{N_2(n)-n\pi_2}{\sqrt n} =-\frac{N_1(n)-n\pi_1}{\sqrt n}. \] Including both count coordinates in a covariance matrix therefore produces a singular matrix by construction. This is not a failure of the limit theorem; it is a deterministic constraint.

Diagnostics, Alternatives, and Limitations

Before using the theorem, check that the assumptions match the design and that the resulting covariance is coherent. In particular, verify that every \(\pi_i\) is positive, the allocation-count representation holds at the square-root-\(n\) scale, and the joint conditional covariance sums and Lindeberg condition are justified. The full block matrix must be positive semidefinite; a negative eigenvalue beyond numerical tolerance can indicate an incorrect cross-covariance, an inconsistent estimated covariance block, or a violated assumption.

A common mistake is to calculate separate marginal limits and then set the cross-covariance to zero because the response innovations have conditional mean zero. That argument accounts for the immediate response-score increment, but not necessarily for the whole adaptive allocation-count fluctuation: future allocation probabilities may depend on earlier responses. The correction is to derive or estimate the joint block \(C\) under the actual allocation-count representation. Zero cross-covariance is justified in settings such as the fixed-probability example, not by conditional centering alone.

The main alternative is to report separate marginal asymptotic distributions. That can be adequate when only one margin is needed, but it is insufficient for joint inference or summaries combining allocation and response estimates. Conditioning on the realized counts is another possible analysis, but it answers a conditional question and does not automatically reproduce the unconditional joint law under adaptive assignment.

Finally, the theorem is not automatic for every adaptive design. If allocation fluctuations do not have the stated square-root-\(n\) representation, if some allocation fractions approach zero, or if conditional covariance limits remain random, the displayed deterministic Gaussian covariance need not apply. Such cases require a different normalization or a limit statement that retains the random conditional covariance. A correct-looking covariance calculation cannot compensate for an unsuitable asymptotic regime.