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

arXiv: https://arxiv.org/abs/2405.13798 · HTML: https://mlsanity.com/paper/2405.13798/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 54 references that are in the map.
- First posted on arXiv 22 May 2024.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [Extracting Training Data from Large Language Models](https://mlsanity.com/paper/2012.07805/) (2020)
- [DetectGPT: Zero-Shot Machine-Generated Text Detection using Probability Curvature](https://mlsanity.com/paper/2301.11305/) (2023)
- [GLTR: Statistical Detection and Visualization of Generated Text](https://mlsanity.com/paper/1906.04043/) (2019)
- [Spotting LLMs With Binoculars: Zero-Shot Detection of Machine-Generated Text](https://mlsanity.com/paper/2401.12070/) (2024)
- [Artificial Fingerprinting for Generative Models: Rooting Deepfake Attribution in Training Data](https://mlsanity.com/paper/2007.08457/) (2020)
- [The Science of Detecting LLM-Generated Texts](https://mlsanity.com/paper/2303.07205/) (2023)
- [A Survey on LLM-Generated Text Detection: Necessity, Methods, and Future Directions](https://mlsanity.com/paper/2310.14724/) (2023)
- [Language Model Evaluation Beyond Perplexity](https://mlsanity.com/paper/2106.00085/) (2021)
- [Position: Understanding LLMs Requires More Than Statistical Generalization](https://mlsanity.com/paper/2405.01964/) (2024)
- [A Tale of Two Perplexities: Sensitivity of Neural Language Models to Lexical Retrieval Deficits in Dementia of the Alzheimer's Type](https://mlsanity.com/paper/2005.03593/) (2020)

## Built on by (most cited first, 4 of 4)

- [LLM Probability Concentration: How Alignment Shrinks the Generative Horizon](https://mlsanity.com/paper/2506.17871/) (2025)
- [Inv-Entropy: A Fully Probabilistic Framework for Uncertainty Quantification in Language Models](https://mlsanity.com/paper/2506.09684/) (2025)
- [Not All Denoising Steps Are Equal: Model Scheduling for Faster Masked Diffusion Language Models](https://mlsanity.com/paper/2604.02340/) (2026)
- [Rethinking Probability-Based Reinforcement Learning From Posterior Concentration](https://mlsanity.com/paper/2610.01458/) (2026)

## Abstract (arXiv)

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.