Description
- Abstract:
- The study of high-dimensional measures is an integral part of probability, statistics and asymptotic convex geometry, where the object of interest is the normalized volume measures of a convex body. A common theme is to try to analyze these high-dimensional objects by looking at their lower-dimension projections. Classical theorems in probability theory on independent and identically distributed random variables, such as the law of large numbers and central limit theorem, provide information about scalar projections of high-dimensional product measures along a particular direction (i.e., the direction $(1, \dots, 1)$). Although the study of (typical) projections of more general (non-product) high-dimensional measures dates back to Borel, only recently has a theory begun to emerge in the field of asymptotic convex geometry, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. In this thesis, we study large deviation properties for random projections of high-dimensional measures and their applications to asymptotic convex geometry. The thesis is divided into three parts as follows. In the first part of the thesis, we study large deviation principles for random multidimensional projections, which are averaged over the randomness of the projections (the so-called annealed setting). Antilla et al. showed that if a random vector $X^n\in\mathbb{R}^n$ satisfies a {\it thin-shell} condition, which is a quantitative estimate on how the mass is concentrated around a thin sphere, then for most directions $\theta^n$, fluctuations of most lower-dimension projections $\langle X^n,\theta^n \rangle$ are almost Gaussian. We introduce an {\it asymptotic thin-shell condition}, which describes the asymptotic rate of concentration of the mass around a thin sphere as the dimension grows to infinity, and show that this condition is sufficient for annealed random projections of the high-dimensional vector to satisfy a large deviation principle. We also verify the asymptotic thin-shell condition for several classes of high-dimensional measures including normalized volume measures of a large class of Orlicz balls. In the second part of the thesis, we obtain refined sharp large deviation estimates for annealed random projections under an additional asymptotic density condition and apply these large deviation estimates to study problems of interest in asymptotic convex geometry. The study of intersections of convex bodies was originated by a classical question raised by Milman, ``What is the volume left in a unit $\ell^n_p$ ball after removing a $t$-multiple of $\ell^n_q$ ball?" To answer this question, Schechtman and Schmuckenschl{\"a}ger studied the limit of the ratio of $\abs{B^n_{p}(1)\cap B^n_{q}(t)}$ and $\abs{B^n_{p}(1)}$, where $B^n_p(t)$ is an $\ell^n_p$ ball with radius $t$. They showed that an interesting phase transition occurs in $t$: the ratio converges to either $0$ or $1$ in a certain {\it subcritical} and {\it supercritical} regime, respectively. The critical regime is more subtle and was finally resolved after a decade by Schechtman, who showed the ratio converges to $1/2$ as $n\to\infty$. The corresponding results for other classes of convex bodies were not as well understood. We extend these results to Orlicz balls and obtain again a phase transition by deriving a sharp estimate for the volume of the intersection of two Orlicz balls. Various other volumetric properties for Orlicz balls are also obtained. In the last part of the thesis, we study quenched (i.e., conditioned on the random projection bases) large deviation estimates for random multidimensional projections, and investigate what geometric information they capture about the high-dimensional measures. LDPs for random projections of $\ell^n_p$ balls were obtained in Gantert et al. While they captured some geometric information, they could not distinguish between balls and spheres. Sharp large deviation estimates were originated obtained for sums of independent and identically distributed random variables $\{X_i\}_{i\in\mathbb{N}}$ by Bahadur and Ranga Rao. With the perspective that such a sum can be viewed as the projection of a product measure in the $(1,...,1)$ direction, we generalize these results in two ways. First, we look at projections along a random fixed direction $\theta^n\in \mathbb{S}^{n-1}$ sampled uniformly from the unique rotational invariant measure on $\mathbb{S}^{n-1}$. Next, we consider random vectors $X^n\in\mathbb{R}^n$ distributed according to some non-product measures. The main idea behind the derivation of the corresponding sharp large deviation estimates is to identify the correct change of measure for the rare event corresponding to the deviation and to obtain a quantitative central limit theorem under this change of measure. We also propose an importance sampling algorithm to numerically approximate the tail probability or the volume of spherical caps of $\ell^n_p$ balls. Numerical simulations show that the sharp large deviation estimate is a very good approximation even when the dimension of the vector is not particularly high.
- Notes:
- Thesis (Ph. D.)--Brown University, 2022
Citation
Liao, Yin-Ting,
"Sharp large deviation estimates and their applications to asymptotic convex geometry"
(2022).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/te1f-0245
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....