Slaves to the Law of Large Numbers: An Asymptotic Equipartition Property for Perplexity in Generative Language Models

We prove a new asymptotic un-equipartition property for the perplexity of long texts generated by a language model and present supporting experimental evidence from open-source models. Specifically we show that the logarithmic perplexity of any large text generated by a language model must asymptotically converge to the average entropy of its token distributions. This defines a ``typical set'' that all long synthetic texts generated by a language model must belong to. We refine the concept of ''typical set'' to include only grammatically correct texts. We then show that this refined typical set is a vanishingly small subset of all possible grammatically correct texts for a very general definition of grammar. This means that language models are strongly constrained in the range of their possible behaviors and outputs. We make no simplifying assumptions (such as stationarity) about the statistics of language model outputs, and therefore our results are directly applicable to practical real-world models without any approximations. We discuss possible applications of the typical set concept to problems such as detecting synthetic texts and membership inference in training datasets.