NSF-CBMS Research Conference Small Deviation Probabilities: Theory and Applications

Ten Lectures on Small Deviation Probabilities: Theorey and Applications Wenbo V. Li (University of Delaware) Lecture 1: Introduction, overview and applications. We first define the small deviation (value) probability in several settings, which basically study the asymptotic rate of approaching zero for rare events that positive random variables take smaller values. Many applications discussed in the scientific justification section are given. Benefits and differences of various formulations of small deviation probabilities are examined in details, together with connections to related fields. Lecture 2: Basic estimates and equivalent transformations. We first formulate several equivalent results for small deviation probability, including negative moments, exponential moments, Laplace transform and Taubirean theorems. The basic techniques involved are various useful inequalities, motivated from large deviation estimates. Some refinement of known results are given, including the classical Paley-Zegmund inequality. Applications to regularity and smoothness of probability laws via small deviation estimates of the determinant of Malliavin matrix are discussed in the setting of stochastic (partial) differential equations. Lecture 3: Techniques associated with independent variables. We start with probabilistic arguments for algebraic properties of small deviation probability, such as independent sums and products. These estimates are non-asymptotic and hence they can be applied in the setting of conditional probability. Separate treatments are analyzed for exponential and power decay rates. A newly discover symmetrization inequality is proved by Fourier analytic method. Littlewood and Offord type problems are discussed. We end with Komlos Conjecture on balancing vectors in discrepancy theory. Lecture 4: Blocking techniques for the sup-norm. We first present the vary useful blocking techniques for the maximum of the absolute value of partial sums in both upper and lower bound setting. The lower bound is more involved since the end position of each block has to be controlled also. The resulting estimates play a critical role in the Chung’s type strong limit theorems for sample paths. Similar techniques are applied to weighted and/or controlled sup-norms for Brownian motion and stable processes. Applications to the two-sided exit time and Wichura type functional limit theorems are indicated. Lecture 5: Links between small ball probabilities and metric entropy. For a continuous centered Gaussian process, the generating linear operator is compact and so is the unit ball of the associated reproducing kernel Hilbert space. The fundamental links between small ball probability for Gaussian measure and the metric entropy are given and various far-reaching implications are explored. Several purely probabilistic results, obtained via the analytic connection without direct probabilistic proofs, are analyzed. Lecture 6: Small deviation (ball) estimates for sums of correlated Gaussian elements. We treat the sum of two not necessarily independent Gaussian random