Projections of Probability Distributions: A Measure-Theoretic Dvoretzky Theorem

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Result states that under isotropic and mild shell/concentration assumptions one-dimensional and k-dimensional random projections converge to Gaussian for k up to about 2 log d / log log d, with bounded‑Lipschitz error →0. Proof uses Stein's method with exchangeable pairs, concentration on the Stiefel manifold and Dudley entropy bounds to control the supremum over Lipschitz tests. A concrete simplex counterexample shows the Gaussian approximation fails once k exceeds that constant‑times log threshold.

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1. 0:00Intro
2. 1:06Marginals are normally Gaussian
3. 8:23Higher-dimensional marginals
4. 18:11Sharpness
5. 22:24Dvoretzky's Theorem
6. 24:08The analogy
7. 24:55Dvoretzky dimension
8. 27:44Outline of the proof of the main theorem
9. 31:20More about step 1
10. 35:40Concentration of measure
11. 37:37Step 2 - Average distance to average