The Annals of Probability

Large deviations for random projections of $\ell^{p}$ balls

Nina Gantert, Steven Soojin Kim, and Kavita Ramanan

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Abstract

Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\ell^{p}$ ball in $\mathbb{R}^{n}$ 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 $\infty$, 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 $p\in(1,\infty]$ (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 Cramér’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 $\ell^{p}$ balls.

Article information

Source
Ann. Probab., Volume 45, Number 6B (2017), 4419-4476.

Dates
Received: December 2015
Revised: November 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1513069264

Digital Object Identifier
doi:10.1214/16-AOP1169

Mathematical Reviews number (MathSciNet)
MR3737915

Zentralblatt MATH identifier
06838124

Subjects
Primary: 60F10: Large deviations
Secondary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 60K37: Processes in random environments 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Large deviations random projections $\ell^{p}$-balls annealed and quenched large deviations self-normalized large deviations central limit theorem for convex sets variational formula

Citation

Gantert, Nina; Kim, Steven Soojin; Ramanan, Kavita. Large deviations for random projections of $\ell^{p}$ balls. Ann. Probab. 45 (2017), no. 6B, 4419--4476. doi:10.1214/16-AOP1169. https://projecteuclid.org/euclid.aop/1513069264


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