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The large and moderate deviations approach in geometric functional analysis

Joscha Prochno

Abstract

The work of Gantert, Kim, and Ramanan [Large deviations for random projections of $\ell^p$ balls, Ann. Probab. 45 (6B), 2017] has initiated and inspired a new direction of research in the asymptotic theory of geometric functional analysis. The moderate deviations perspective, describing the asymptotic behavior between the scale of a central limit theorem and a large deviations principle, was later added by Kabluchko, Prochno, and Thäle in [High-dimensional limit theorems for random vectors in $\ell_p^n$ balls. II, Commun. Contemp. Math. 23(3), 2021]. These two approaches nicely complement the classical study of central limit phenomena or non-asymptotic concentration bounds for high-dimensional random geometric quantities. Beyond studying large and moderate deviations principles for random geometric quantities that appear in geometric functional analysis, other ideas emerged from the theory of large deviations and the closely related field of statistical mechanics, and have provided new insight and become the origin for new developments. Within less than a decade, a variety of results have appeared and formed this direction of research. Recently, a connection to the famous Kannan-Lovász-Simonovits conjecture and the study of moderate and large deviations for isotropic log-concave random vectors was discovered. In this manuscript, we introduce the basic principles, survey the work that has been done, and aim to manifest this direction of research, at the same time making it more accessible to a wider community of researchers.

The large and moderate deviations approach in geometric functional analysis

Abstract

The work of Gantert, Kim, and Ramanan [Large deviations for random projections of balls, Ann. Probab. 45 (6B), 2017] has initiated and inspired a new direction of research in the asymptotic theory of geometric functional analysis. The moderate deviations perspective, describing the asymptotic behavior between the scale of a central limit theorem and a large deviations principle, was later added by Kabluchko, Prochno, and Thäle in [High-dimensional limit theorems for random vectors in balls. II, Commun. Contemp. Math. 23(3), 2021]. These two approaches nicely complement the classical study of central limit phenomena or non-asymptotic concentration bounds for high-dimensional random geometric quantities. Beyond studying large and moderate deviations principles for random geometric quantities that appear in geometric functional analysis, other ideas emerged from the theory of large deviations and the closely related field of statistical mechanics, and have provided new insight and become the origin for new developments. Within less than a decade, a variety of results have appeared and formed this direction of research. Recently, a connection to the famous Kannan-Lovász-Simonovits conjecture and the study of moderate and large deviations for isotropic log-concave random vectors was discovered. In this manuscript, we introduce the basic principles, survey the work that has been done, and aim to manifest this direction of research, at the same time making it more accessible to a wider community of researchers.
Paper Structure (37 sections, 27 theorems, 234 equations, 2 figures)

This paper contains 37 sections, 27 theorems, 234 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The large and moderate deviations perspective
  3. An outrageously short introduction to large deviations theory
  4. Motivation
  5. Cramér type large deviations
  6. Sanov's theorem for the empirical measure
  7. The Gärtner--Ellis theorem for dependent random variables
  8. The abstract theory of large deviations
  9. Three fundamental principles
  10. Large deviations results and techniques in the world of $\ell_p^n$-spaces
  11. Large deviations for random projections of $\ell_p^n$-balls
  12. Annealed LDP -- the case $p\in[2,\infty)$
  13. Annealed LDP -- the case $p\in[1,2)$
  14. Large deviations for $\ell_q$-norms of random vectors in $\ell_p$-balls
  15. Moderate deviations for $\ell_q$-norms of random vectors in $\ell_p$-balls
  16. ...and 22 more sections

Key Result

Theorem 2.1

Let $X_1,X_2,\dots$ be iid random variables such that, for all $t\in\mathbb R$, Then, for every $a>\mathbb E[X_1]$, where $\Lambda_X^*(a):=\sup_{s\in\mathbb R}[as-\Lambda_X(s)]$ is the Legendre transform of $\Lambda_X$.

Figures (2)

  • Figure 1: Plots of the densities $\frac{\mu_{\infty}^{(1)}(\textup{d} x)}{\textup{d} x}$ (left) and $\frac{\mu_{\infty}^{(2)}(\textup{d} x)}{\textup{d} x}$ (right).
  • Figure 2: Plots of the densities $\frac{\eta_{\infty}^{(1)}(\textup{d} x)}{\textup{d} x}$ (left) and $\frac{\eta_{\infty}^{(2)}(\textup{d} x)}{\textup{d} x}$ (right).

Theorems & Definitions (60)

  • Theorem 2.1: H. Cramér, 1938
  • proof : Idea of Proof.
  • Theorem 2.2: Cramér for stretched exponential tails
  • proof : Idea of Proof of Theorem \ref{['thm:cramer stretched']}.
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5: Sanov's theorem
  • Theorem 2.6: Gärtner--Ellis theorem
  • Definition 2.7: Rate function and LDP
  • Remark 2.8
  • ...and 50 more