The method of Laplace transformation is also known as Tauberian theorem or time-exponentiation, and has been applied to a variety of the problems associated to Brownian motions (and other models) such as small ball probabilities, large deviations, moment computation of local and intersection local times, and ray-knight theorem. In this talk, examples will be given to demonstrate how this powerful tool is used in different settings and related questions will be asked.
In this talk, I will present a number of examples/problems in convex geometry and in bracketing entropy where small ball probability is a powerful tool, and examples where tools from these areas are useful to estimate small ball probability.
The classical central limit theorem in probability theory states that if \(X_1, \cdots, X_n\) are iid with mean \( \mu \), then (under some mild conditions)
\[F_n = \sqrt{n} \left( \frac{X_1 + \cdots + X_n}{n} - \mu\right) \]converges in distribution to a normal distribution. This is true for many other random variables \( F_n \) such as multiple Wiener-Ito integrals. In this talk we shall discuss under what condition, \( F_n \) have densities and the densities of \( F_n \) converge to the normal density. More precisely, we will consider the problem of finding conditions such that there are integrable positive functions \( f_n(x) \) such that
\[ P(a \le F_n \le b) = \int_a^b f_n(x) dx \]and
\[ \lim_{n \rightarrow \infty} \int_{-\infty}^\infty |f_n(x) - \phi(x) |^p dx = 0 \]
for all \(p \ge 1\), where \( \phi(x) = \frac1{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \) is the density of standard normal. The tool that we use is the Malliavin calculus. This is an ongoing joint work with Fei Lu and David Nualart.
We establish empirical quantile process CLT's based on \(n\) independent copies of a stochastic process \(\{X_t: t \in E \} \) that are uniform in \(t \in E \) and quantile levels \( \alpha \in I \), where \(I\) is a closed sub-interval of \( (0,1) \). Typically \( E = [0,T] \), or a finite product of such intervals. Also included are CLT's for the empirical process based on \( \{I_{X_t \le y} - P(X_t \le y): t \in E, \; y \in \R \} \) that are uniform in \(t \in E, y \in \R \). The process \( \{X_t: t \in E\} \) may be chosen from a broad collection of Gaussian processes, compound Poisson processes, stationary independent increment stable processes, and martingales.
In the first part of the talk I will give a short introduction into learning theory, in order to show the importance of metric entropy in this very active field. A particular problem - which is related, e.g., to support vector machines - consists in finding good upper bounds for entropy numbers (or covering numbers) in reproducing kernel Hilbert spaces. In the second part I will determine the exact asymptotic behaviour of covering numbers in Gaussian RKHSs. On one hand, this improves earlier results by Ding-Xuan Zhou, and on the other hand it has an interpretation in terms of small deviations of certain smooth Gaussian processes. This part is based on my paper Covering numbers in Gaussian reproducing kernel Hilbert spaces
J. Complexity 27 (2011), 489--499.
We will survey the remarkably close connection with small ball problems in probability theory, and the classical bounds associated with the Discrepancy function. Tools to analysize these questions arise from Harmonic Analysis, but don't seem strong enough to complete the proofs of outstanding conjectures in the subject.
Smoothness properties of measures in infinite-dimensional spaces, in particular the laws of Brownian motions in these spaces, have been the subject of much research in various settings, including certain curved examples. We will consider the setting of infinite-dimensional Heisenberg-like
groups. The Brownian motions in this case may be realized as a flat infinite-dimensional Brownian motion along with its second-order chaos. We will discuss recent smoothness results for the law of these processes; reverse log Sobolev inequalities play a critical role in the proof of these results. I will provide all basic definitions as well as some background to put these results in context. This is joint work with F. Baudoin and M. Gordina.
In the first half of the talk I will introduce several versions of the Erdos and Littlewood-Offord inequality. In the second half I will then present an application to bound the least singular value and to establish the circular law in random matrix theory.
Small ball probabilities are very important for investigating fine structires of the sample functions of Gaussian random fields. In this talk we present applications of small ball probabilities in establishing results on exact Hausdorff measure functions, exact packing measure functions and multifractal analysis for Gaussian random fields.