Daniel Raban's Note Repository

Here are some notes I've written. If you find any typos or inaccuracies in these notes, or if you have any suggestions on how to improve the notes (aesthetically or otherwise), please send me an email at ude.yelekreb@nabarleinad. I appreciate the help!

I am a statistics PhD candidate at UC Berkeley, advised by Steve Evans. For the 2025-2026 academic year, I am a visiting student at Yale University. I received my BA in math from UC Berkeley and my MA in math from UCLA. My research interests span a variety of topics in probability and statistics, such as optimal transport, high-dimensional statistics, ergodic theory, game theory, stochastic processes, and the analysis of ratings.

Why the name Pillowmath? Because it's comfy!

Misc.

• The Problem With Averaging Ratings (And How to Fix It)
• Musings About Video Game Music

Papers

• Coupling Theory, Optimal Transport, and Strassen's Theorem Beyond Regular Orders (with Adam Quinn Jaffe, 2025)
• Laws of Large Numbers for Information Resolution (2025)
• How should we aggregate ratings? Accounting for personal rating scales via Wasserstein barycenters (2024)
  • For a brief, non-technical overview, here's a 3-minute talk I gave on this as a UC Grad Slam semi-finalist.
  • I also wrote a short (8-minute) article for a general audience.
  • Here's Google NotebookLM explaining this paper as a short podcast.
• Rotatable Random Sequences in Local Fields (with Steven N. Evans, arXiv version)

Electronic Communications in Probability 24, (2019)

Expository Notes

• Measure Theory for Analysts and Probabilists (Updated 7/16/18)
• Representation Theory for Mortals (Updated 7/12/19)
• Fixed Point Theorems and the Existence of Haar Measure
• The Glivenko-Cantelli Theorem and Introduction to VC Dimension
• Topological Models of Measure-Preserving Systems (Updated 3/4/21)
• High-Dimensional Statistics and the Johnson-Lindenstrauss Embedding
• Ergodic Theorems for Spherical Averages of Free Group Actions

Course Notes