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Gaussian universality for approximately polynomial functions of high-dimensional data
- Publication Year :
- 2024
-
Abstract
- We establish an invariance principle for polynomial functions of $n$ independent high-dimensional random vectors, and also show that the obtained rates are nearly optimal. Both the dimension of the vectors and the degree of the polynomial are permitted to grow with $n$. Specifically, we obtain a finite sample upper bound for the error of approximation by a polynomial of Gaussians, measured in Kolmogorov distance, and extend it to functions that are approximately polynomial in a mean squared error sense. We give a corresponding lower bound that shows the invariance principle holds up to polynomial degree $o(\log n)$. The proof is constructive and adapts an asymmetrisation argument due to V. V. Senatov. As applications, we obtain a higher-order delta method with possibly non-Gaussian limits, and generalise a number of known results on high-dimensional and infinite-order U-statistics, and on fluctuations of subgraph counts.
- Subjects :
- Mathematics - Probability
Mathematics - Statistics Theory
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2403.10711
- Document Type :
- Working Paper