On the Role of Tail Behavior in Learning and Inference

Modern data analysis increasingly relies on both artificial intelligence and statistical tools. Distributions provide a concise and powerful language for describing the behavior of data and the methods applied to them. One particularly important aspect of a distribution is its tail behavior. This dissertation studies tail behavior as a governing quantity in two settings: post-training for large language models and p-value aggregation in statistical inference. In both settings, we show that exploiting the tail properties of relevant distributions leads to sharper theoretical understanding and improved methodology.

In the first part, we study LLM post-training, where the goal is to align a language model toward responses with high rewards. Such responses are rare under the base model and reside in the tail of its induced reward distribution. In Chapter 2, we show that the best-of-n sampling distribution, which targets the upper tail of the reward distribution, is essentially optimal for maximizing win rate against the base model at each level of KL divergence. Motivated, we develop BoNBoN, an effective method for training a language model to mimic this distribution. In Chapter 3, we demonstrate that the accuracy of the reward model in the high-reward tail is the primary determinant of post-training quality: mis-specification in this region drives reward over-optimization. Motivated by this finding, we develop a rubric-based reward modeling approach that constructs rewards by distinguishing among the best responses, and empirical evidence supprts its efficacy.

In the second part, we study heavy-tailed combination tests for testing global null hypotheses. In Chapter 4, we show that under asymptotic independence of test statistics, heavy-tailed combination tests are asymptotically valid yet asymptotically equivalent to the Bonferroni test. This equivalence follows from a fundamental univariate tail property of regularly varying distributions: the tail of the sum is governed by the maximum. We further provide empirical evidence that these tests can outperform Bonferroni under asymptotic dependence. In Chapter 5, we develop a broader theoretical framework based on multivariate regularly varying copulas, which characterize the joint tail dependence structure among p-values near zero. Within this framework, we establish that heavy-tailed combination tests with tail index $\gamma\le1$ are asymptotically valid across a wide class of dependence structures, and that their power advantage over the Bonferroni test grows monotonically with the strength of tail dependence.