Georgia Institute of Technology College of Sciences

The focus of my talk will be stochastic forms of isoperimetric inequalities for convex sets. I will review some fundamental inequalities including the classical isoperimetric inequality and those of Brunn-Minkowski and Blaschke-Santalo on the product of volumes of a convex body and its polar dual. I will show how one can view these as global inequalities that arise via random approximation procedures in which stochastic dominance holds at each stage. By laws of large numbers, these randomized versions recover the classical inequalities. I will discuss when such stochastic dominance arises and its applications in convex geometry and probability. The talk will be expository and based on several joint works with G. Paouris, D. Cordero-Erausquin, M. Fradelizi, S. Dann and G. Livshyts.

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities Pr(n−1/2∑ni=1Xi∈A) where X1,…,Xn are independent random vectors in ℝp and Ais a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn→∞ as n→∞ and p≫n; in particular, p can be as large as O(eCnc) for some constants c,C>0. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

We prove a central limit theorem for a class of additive processes that arise naturally in the theory of finite horizon Markov decision problems. The main theorem generalizes a classic result of Dobrushin (1956) for temporally non-homogeneous Markov chains, and the principal innovation is that here the summands are permitted to depend on both the current state and a bounded number of future states of the chain. We show through several examples that this added flexibility gives one a direct path to asymptotic normality of the optimal total reward of finite horizon Markov decision problems. The same examples also explain why such results are not easily obtained by alternative Markovian techniques such as enlargement of the state space. (Joint work with J. M. Steele.)