When Collapsing Exploration Breaks the Central Limit Theorem

Why a Vanishing Allocation Fraction Is Not the Whole Story

In an adaptive trial, exploration may become less frequent as the trial proceeds. A shrinking allocation probability does not by itself invalidate a Gaussian approximation for an arm mean. What matters is how many observations the arm still accumulates, and whether any individual observation can dominate the total fluctuation. If exploration collapses too quickly, the arm mean may remain based on only a few observations—or be undefined because the arm was never assigned.

To isolate this issue, begin with a two-arm design in which arm A is assigned to exactly \(m_n\) of the \(n\) participants. Its allocation fraction is \(p_n=m_n/n\). Suppose the observed arm-A responses are independent and identically distributed with mean \(\mu_A\) and finite, positive variance \(\sigma_A^2\). This fixed-quota setup makes the information available for estimating \(\mu_A\) explicit. In an adaptive design, the corresponding quantity is the realized or accumulated exposure to arm A; additional conditions are then needed to handle its dependence on past data.

The Gaussian Benchmark and Its Conditions

For a fixed quota \(m_n\), write the arm-A responses as \(Y_{n1},\ldots,Y_{nm_n}\), and let \(\bar{Y}_{A,n}\) be their mean. When \(m_n\) tends to infinity and the responses have a common distribution with finite, positive variance, the ordinary central limit theorem gives

Thus the allocation fraction \(p_n\) may tend to zero while the Gaussian limit remains valid: it is enough in this benchmark that \(np_n=m_n\) tends to infinity. The rate of convergence can be slow, and a finite sample may still be poorly approximated by a normal distribution, but the asymptotic averaging mechanism is present.

For independent, centered variables \(X_{n1},\ldots,X_{nk_n}\), define the total variance \(s_n^2=\sum_j\operatorname{Var}(X_{nj})\). The Lindeberg condition is

The Lindeberg–Feller central limit theorem states that, under independence and \(s_n^2>0\), this condition implies \(\sum_jX_{nj}/s_n\xrightarrow{d}N(0,1)\). It formalizes the requirement that no small collection of terms contributes a non-negligible share of total variance. For identically distributed responses with finite variance, the condition follows as \(m_n\to\infty\): the threshold \(\varepsilon\sigma_A\sqrt{m_n}\) tends to infinity, and the expected squared response beyond that threshold tends to zero.

Worked Examples: Three Ways Exploration Can Behave

What Fails When the Count Stays Bounded?

When \(m_n\) stays bounded, the usual \(\sqrt{m_n}\) averaging mechanism never develops. If \(m_n\) is a fixed positive integer, the distribution of the arm mean remains the distribution of an average of that fixed number of outcomes; it does not approach a normal distribution merely because the total trial size \(n\) grows. If \(m_n=0\) in some trials, the mean is undefined. With random assignment probabilities, a useful first diagnostic is the expected count \(np_n\), but that expectation alone does not describe the variability of the realized count.

For example, if \(p_n=2/n\), then \(N_{A,n}\) converges in distribution to a Poisson random variable with mean 2. The probability of no arm-A assignment tends to \(e^{-2}\), not zero. Even in trials where arm A is observed, the number of responses does not grow without bound in probability. A Gaussian approximation to the arm mean cannot be justified by a large total trial size when the arm-specific sample size remains of this order.

There is a related distinction between the variance of a sum and the validity of a normal approximation. A centered sum can be divided by its own standard deviation and have variance one for every \(n\), yet still fail to be asymptotically normal. For instance, with a very sparse random assignment, the standardized sum may be zero with probability approaching one and very large on rare assignment events. Variance standardization does not erase that concentration of probability. The Lindeberg condition checks precisely whether such rare, dominant contributions are disappearing.

A Short Justification for the Fixed-Distribution Benchmark

Suppose the arm-A responses have a common distribution with mean \(\mu_A\) and variance \(0<\sigma_A^2<\infty\), and \(m_n\to\infty\). Take \(X_{nj}=Y_{nj}-\mu_A\). Then \(s_n^2=m_n\sigma_A^2\). For any \(\varepsilon>0\), the Lindeberg expression becomes

The squared centered response has finite expectation, while the threshold tends to infinity. The indicator therefore tends pointwise to zero, and the integrand is bounded above by the integrable variable \((Y_{n1}-\mu_A)^2\). The dominated convergence theorem makes the displayed expression tend to zero. Lindeberg–Feller then gives the Gaussian limit. This argument also identifies the boundary of the claim: finite variance and a growing count are sufficient in this benchmark, but if the response law changes with \(n\), finite variance in each row does not itself make the Lindeberg expression vanish.

Diagnostics, Alternatives, and Limitations

When assessing a proposed Gaussian arm-mean approximation, examine the arm-specific information rather than the total enrollment alone. In a fixed-probability design, compute \(np_n\); in an adaptive trial, inspect the realized arm count and how it grows across trial sizes. Also check whether the response distribution is stable across \(n\), whether its variance is finite, and—when distributions change—whether large centered outcomes contribute a vanishing fraction of total variance. Simulations can reveal frequent empty-arm samples or unstable estimates, but a visually plausible histogram at one sample size does not establish an asymptotic CLT.

The main alternative to a collapsing exploration schedule is a design constraint that preserves a minimum amount of exploration, such as a fixed positive allocation floor or a minimum arm-specific quota. This can support growing arm counts, but it does not by itself repair heavy tails or a changing response distribution that violates Lindeberg’s condition. When arm counts are small or outcomes have influential tails, exact or simulation-based procedures tailored to the design may be preferable to a normal approximation; they still require their own assumptions and calibration.

For adaptive allocation, the fixed-quota argument is a benchmark, not a complete theorem. Allocation decisions may depend on past outcomes, and the arm count may be random. A Gaussian limit then needs conditions that control both the accumulated conditional variance and large increments. In particular, a small expected allocation probability should not be substituted mechanically for a guarantee about realized information. The central diagnostic remains: does the arm accumulate enough observations, and can any one contribution remain a non-negligible part of the normalized sum?