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A conditional limit theorem for high-dimensional ℓᵖ-spheres

Published online by Cambridge University Press:  16 January 2019

Steven S. Kim*
Affiliation:
Brown University
Kavita Ramanan*
Affiliation:
Brown University
*
* Postal address: Brown University, 182 George Street, Box F, Providence, RI 02912, USA.
* Postal address: Brown University, 182 George Street, Box F, Providence, RI 02912, USA.

Abstract

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Alonso-Gutiérrez, D., Prochno, J. and Thäle, C. (2018). Large deviations for high-dimensional random projections of ℓpn balls. Adv. Appl. Math. 99, 135.CrossRefGoogle Scholar
[2]Ambrosio, L., Gigli, N. and Savaré, G. (2008). Gradient flows in metric spaces and in the space of probability measures (Lectures Math. ETH Z\"urich), 2nd edn. Birkhäuser, Basel.Google Scholar
[3]Barthe, F., Guédon, O., Mendelson, S. and Naor, A. (2005). A probabilistic approach to the geometry of the ℓpn-ball. Ann. Prob. 33, 480513.CrossRefGoogle Scholar
[4]Ben Arous, G., Dembo, A. and Guionnet, A. (2001). Aging of spherical spin glasses. Prob. Theory Relat. Fields 120, 167.CrossRefGoogle Scholar
[5]Ben Arous, G. and Guionnet, A. (1997). Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Prob. Theory Relat. Fields 108, 517542.CrossRefGoogle Scholar
[6]Bertini, L.et al. (2015). Macroscopic fluctuation theory. Rev. Modern Phys. 87, 593636.CrossRefGoogle Scholar
[7]Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
[8]Boyd, S. and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press.CrossRefGoogle Scholar
[9]Cover, T. M. and Thomas, J. A. (2006). Elements of Information Theory, 2nd edn. John Wiley, Hoboken, NJ.Google Scholar
[10]Csiszár, I. (1984). Sanov property, generalized I-projection and a conditional limit theorem. Ann. Prob. 12, 768793.CrossRefGoogle Scholar
[11]Dellacherie, C. and Meyer, P.-A. (2011). Probabilities and Potential, C (North-Holland Math. Studies 151). North-Holland, Amsterdam.Google Scholar
[12]Dembo, A. and Shao, Q.-M. (1998). Self-normalized large deviations in vector spaces. In High Dimensional Probability (Progr. Prob. 43). Birkhäuser, Basel, pp. 2732.CrossRefGoogle Scholar
[13]Dembo, A. and Zeitouni, O. (1996). Refinements of the Gibbs conditioning principle. Prob. Theory Relat. Fields 104, 114.CrossRefGoogle Scholar
[14]Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38), 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
[15]Diaconis, P. and Freedman, D. (1987). A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Statist. 23, 397423.Google Scholar
[16]Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York.CrossRefGoogle Scholar
[17]Ellis, R. S. (2006). Entropy, Large Deviations, and Statistical Mechanics. Springer, Berlin.CrossRefGoogle Scholar
[18]Gantert, N., Kim, S. S. and Ramanan, K. (2006). Cramér's theorem is atypical. In Advances in the Mathematical Sciences (Assoc. Women Math. Ser. 6). Springer, Cham, pp. 253270.CrossRefGoogle Scholar
[19]Gantert, N., Kim, S. S. and Ramanan, K. (2017). Large deviations for random projections of ℓp balls. Ann. Prob. 45, 44194476.CrossRefGoogle Scholar
[20]Kabluchko, Z., Prochno, J. and Thäle, C. (2017). High-dimensional limit theorems for random vectors in $\ell_p^n$-balls. Commun. Contemp. Math. 1750092.Google Scholar
[21]Kim, S. S. (2017). Problems at the interface of probability and convex geometry: Random projections and constrained processes. Doctoral Thesis, Brown University.Google Scholar
[22]Kim, S. S. and Ramanan, K. (2018). Large deviations on the Stiefel manifold. Preprint.Google Scholar
[23]Klartag, B. (2007). A central limit theorem for convex sets. Invent. Math. 168, 91131.CrossRefGoogle Scholar
[24]Léonard, C. (2010). Entropic projections and dominating points. ESAIM: Prob. Statist. 14, 343381.CrossRefGoogle Scholar
[25]Léonard, C. and Najim, J. (2002). An extension of Sanov's theorem: application to the Gibbs conditioning principle. Bernoulli 8, 721743.Google Scholar
[26]Lynch, J. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Prob. 15, 610627.CrossRefGoogle Scholar
[27]Mogul′skii, A. A. (1991). De Finetti-type results for ℓp. Siberian Math. J. 32, 609619.CrossRefGoogle Scholar
[28]Naor, A. (2007). The surface measure and cone measure on the sphere of ℓpn. Trans. Amer. Math. Soc. 359, 10451079.CrossRefGoogle Scholar
[29]Naor, A.and Romik, D. (2003). Projecting the surface measure of the sphere of ℓpn. Ann. Inst. H. Poincaré Prob. Statist. 39, 241261.CrossRefGoogle Scholar
[30]Rachev, S. T. and Rüschendorf, L. (1991). Approximate independence of distributions on spheres and their stability properties. Ann. Prob. 19, 13111337.CrossRefGoogle Scholar
[31]Schechtman, G. and Zinn, J. (1990). On the volume of the intersection of two L pn balls. Proc. Amer. Math. Soc. 110, 217224.Google Scholar
[32]Spruill, M. C. (2007). Asymptotic distribution of coordinates on high dimensional spheres. Electron. Commun. Prob. 12, 234247.CrossRefGoogle Scholar
[33]Stam, A. J. (1982). Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces. J. Appl. Prob. 19, 221228.CrossRefGoogle Scholar
[34]Stroock, D. W.and Zeitouni, O. (1991). Microcanonical distributions, Gibbs states, and the equivalence of ensembles. In Random Walks, Brownian Motion, and Interacting Particle Systems (Progr. Prob. 28). Birkhäuser, Boston, MA.Google Scholar
[35]Sznitman, A.-S. (1991). Topics in propagation of chaos. In Ecole d’été de probabilités de Saint-Flour XIXXIX—1989 (Lecture Notes Math. 1464). Springer, Berlin, pp. 165251.CrossRefGoogle Scholar
[36]Trashorras, J. (2002). Large deviations for a triangular array of exchangeable random variables. Ann. Inst. H. Poincaré Prob. Statist. 38, 649680.CrossRefGoogle Scholar
[37]Villani, C. (2009). Optimal Transport: Old and New, 338, Springer, Berlin.CrossRefGoogle Scholar
[38]Wang, R., Wang, X. and Wu, L. (2010). Sanov’s theorem in the Wasserstein distance: a necessary and sufficient condition. Statist. Prob. Lett. 80, 505512.CrossRefGoogle Scholar
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