Martingale Central Limit Theorem for Adaptive Designs

Why a Central Limit Theorem Must Allow Adaptation

In an adaptive trial, the probability of assigning the next participant to an arm may depend on the allocation and response history. Consequently, the sequence of observed responses is generally not an independent, identically distributed sample. Yet a useful normal approximation can still hold: after subtracting each response’s conditional mean, the successive contributions can form a martingale difference sequence. A martingale central limit theorem (martingale CLT) is designed for this sequential dependence.

The key checks are not that allocations are independent or that every response has the same marginal distribution. Instead, one checks that the accumulated conditional variance approaches a nonrandom limit and that no single normalized increment contributes appreciably to the sum. The result below is for a deterministic variance limit; it gives a normal approximation for a centered sum, not by itself a limit theorem for a treatment mean with a random denominator.

The Martingale CLT and Its Conditions

For each sample size \(n\), let \(D_{n1},\ldots,D_{nn}\) be martingale differences and let \(s_n>0\) be a nonrandom scaling constant. Define \(X_{ni}=D_{ni}/s_n\). The normalized sum is \(\sum_{i=1}^n X_{ni}\), and its accumulated conditional variance is \(\sum_{i=1}^n E(X_{ni}^2\mid\mathcal{F}_{i-1})\).

The variance condition identifies the scale of the limiting normal distribution. If the conditional variances instead sum to a quantity converging to \(v>0\), normalize the increments by \(\sqrt{v}\) times their current scale, or equivalently apply the theorem after dividing the sum by \(\sqrt{v}\). In this tutorial, the examples use a deterministic limit. A random variance limit requires more care than simply substituting a random number into the ordinary normal approximation.

A short justification explains why the two conditions work together. For a small increment \(X_{ni}\), a conditional Taylor expansion gives \(E(e^{itX_{ni}}\mid\mathcal{F}_{i-1})\) approximately equal to \(1-\tfrac{t^2}{2}E(X_{ni}^2\mid\mathcal{F}_{i-1})\); the linear term vanishes because the conditional mean is zero. The conditional Lindeberg condition makes the accumulated Taylor remainders negligible, including contributions from increments too large for the small-increment approximation. The conditional variance condition makes the accumulated quadratic terms approach \(1\). Thus the characteristic function of the sum approaches \(\exp(-t^2/2)\), the characteristic function of \(N(0,1)\). This is the mechanism behind the theorem, not an assumption that the increments are independent.

Adaptive Response Sums

Consider a trial with treatment arm \(a\). At stage \(i\), let \(A_i=1\) indicate assignment to arm \(a\), and let \(p_i=P(A_i=1\mid\mathcal{F}_{i-1})\) be its predictable assignment probability. Let \(Y_i\) be the observed response. Assume assignment occurs before response observation and, conditional on the past and assignment to arm \(a\), \(E(Y_i\mid\mathcal{F}_{i-1},A_i=1)=\mu_a\) and \(E\{(Y_i-\mu_a)^2\mid\mathcal{F}_{i-1},A_i=1\}=\sigma_a^2\). These assumptions say that the arm-specific conditional mean and variance are stable across stages; they do not require the allocation probabilities \(p_i\) to be constant.

The centered contribution from this arm at stage \(i\) is \(D_i=A_i(Y_i-\mu_a)\). Its conditional expectation is zero: when \(A_i=0\), the contribution is zero, and when \(A_i=1\), the response residual has conditional mean zero. Its conditional variance is \(E(D_i^2\mid\mathcal{F}_{i-1})=p_i\sigma_a^2\). Thus adaptation enters the variance through the predictable probabilities, rather than through an independence assumption.

Suppose \(n^{-1}\sum_{i=1}^n p_i\) converges in probability to a fixed \(q>0\), and \(\sigma_a^2>0\). Then the natural variance scale for the centered sum is \(nq\sigma_a^2\). The corresponding normalized conditional variance is \(\sum_i p_i\sigma_a^2/(nq\sigma_a^2)=\sum_i p_i/(nq)\), which converges in probability to \(1\). Also, \(A_i-p_i\) is a bounded martingale difference. Its average converges to zero, so \(n^{-1}\sum_i A_i\) and \(n^{-1}\sum_i p_i\) have the same probability limit. In particular, \(q\) is the limiting allocation fraction for this arm.

To check negligibility, a convenient sufficient condition is a uniformly bounded conditional \(2+\delta\) moment for some \(\delta>0\): assume \(E(|Y_i-\mu_a|^{2+\delta}\mid\mathcal{F}_{i-1},A_i=1)\le C\), where \(C\) does not depend on \(i\). For \(X_{ni}=D_i/\sqrt{nq\sigma_a^2}\), the inequality \(x^2 I(|x|>\epsilon)\le |x|^{2+\delta}/\epsilon^\delta\) gives

This sufficient bound verifies the theorem’s negligibility condition without requiring a separate calculation for every response value. Bounded response residuals provide another straightforward route: if \(|Y_i-\mu_a|\le B\), then \(|X_{ni}|\le B/\sqrt{nq\sigma_a^2}\), which is eventually less than any fixed \(\epsilon>0\).

A Two-Arm Centered Sum

The same reasoning can check a sum that combines both arms. Suppose the two arms have stable conditional means \(\mu_1,\mu_0\), variances \(\sigma_1^2,\sigma_0^2\), and conditional response residuals with mean zero. Define \(D_i=A_i(Y_i-\mu_1)-(1-A_i)(Y_i-\mu_0)\), where \(A_i=1\) denotes arm 1. Given \(\mathcal{F}_{i-1}\), only one arm contributes, so \(E(D_i\mid\mathcal{F}_{i-1})=0\) and \(E(D_i^2\mid\mathcal{F}_{i-1})=p_i\sigma_1^2+(1-p_i)\sigma_0^2\). If \(n^{-1}\sum_i p_i\) converges to a fixed \(q\), the conditional variance divided by \(n\) converges to \(q\sigma_1^2+(1-q)\sigma_0^2\). If this limit is positive and the increments satisfy the conditional Lindeberg condition after normalization, the centered combined sum, divided by the square root of \(n\) times that limit, converges to \(N(0,1)\).

Diagnostics, Alternatives, and Limitations

A useful verification follows the structure of the theorem rather than relying on a familiar normal-approximation label. First check that each centered contribution has conditional mean zero given the information just before assignment. Then calculate its conditional variance using the current assignment probabilities. Finally, justify that the normalized contributions cannot carry substantial variance in large jumps. For response sums, conditional moment bounds or bounded responses are often practical ways to do the last check.

• Using marginal instead of conditional probabilities. The relevant \(p_i\) is the assignment probability given the history at stage \(i-1\). Replacing it with a single unconditional allocation fraction can give the wrong conditional variance calculation.
• Ignoring a changing response law. If the arm-specific conditional mean or variance changes with the history, the stated formulas need modification. A zero-mean residual alone does not make the conditional variance equal to a fixed \(\sigma_a^2\).
• Assuming a large sample repairs a failed variance condition. The conditional variance sum must converge to the claimed scale. A persistent mismatch, or a variance limit that remains random, is not justified by the theorem as stated.
• Applying the result to a studentized estimator automatically. The theorem concerns a normalized martingale sum with a specified scaling. Random denominators and the limit law of arm-specific means require additional arguments.

An iid central limit theorem is a poor default when the treatment assignment depends on the observed history: its independence premise may fail. A martingale CLT instead uses the sequential conditional mean and variance. An independent triangular-array CLT can handle some changing distributions, but it still does not by itself accommodate history-dependent assignment. The martingale result also has limits: it does not guarantee a normal approximation when one arm is sampled too rarely, when the conditional variance does not stabilize at a positive constant, or when large increments retain non-negligible variance.