Open Access
November 2017 Large deviations for random projections of $\ell^{p}$ balls
Nina Gantert, Steven Soojin Kim, Kavita Ramanan
Ann. Probab. 45(6B): 4419-4476 (November 2017). DOI: 10.1214/16-AOP1169


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.


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Nina Gantert. Steven Soojin Kim. Kavita Ramanan. "Large deviations for random projections of $\ell^{p}$ balls." Ann. Probab. 45 (6B) 4419 - 4476, November 2017.


Received: 1 December 2015; Revised: 1 November 2016; Published: November 2017
First available in Project Euclid: 12 December 2017

zbMATH: 06838124
MathSciNet: MR3737915
Digital Object Identifier: 10.1214/16-AOP1169

Primary: 60F10
Secondary: 52A20 , 60D05 , 60K37

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

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 6B • November 2017
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