LARGE DEVIATIONS FOR RANDOM PROJECTIONS OF lp BALLS

• 成果类型: Article
• 署名作者: Gantert, Nina; Kim, Steven Soojin; Ramanan, Kavita
• 署名单位: Technical University of Munich; Brown University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1169
• 发表日期: 2017
• 页码: 4419-4476
• 关键词: exchangeable random-variables; Moderate Deviations; Empirical Processes; random environment; high dimension; random-walks; convex-sets; distributions; STABILITY; THEOREM

摘要

Let p is an element of[ 1,infinity]. Consider the projection of a uniform random vector from a suitably normalized l(p) ball in R-n onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension n goes to infinity, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for p is an element of (1,infinity] ( but not for p = 1), the corresponding rate function is universal, in the sense that it coincides for almost every sequence of projection directions. We also analyze some exceptional sequences of directions in the measure zero set, including the sequence of directions corresponding to the classical Cramer's theorem, and show that those sequences of directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of l(p) balls.