Gaussian fluctuations for high-dimensional random projections of \(\ell_p^n\)-balls. (English) Zbl 1431.60011

Summary: In this paper, we study high-dimensional random projections of \(\ell_p^n\)-balls. More precisely, for any \(n\in\mathbb{N}\) let \(E_n\) be a random subspace of dimension \(k_n\in\{1,\ldots,n\}\) and \(X_n\) be a random point in the unit ball of \(\ell_p^n\). Our work provides a description of the Gaussian fluctuations of the Euclidean norm \(\|P_{E_n}X_n\|_2\) of random orthogonal projections of \(X_n\) onto \(E_n\). In particular, under the condition that \(k_n\to\infty\) it is shown that these random variables satisfy a central limit theorem, as the space dimension \(n\) tends to infinity. Moreover, if \(k_n\to\infty\) fast enough, we provide a Berry-Esseen bound on the rate of convergence in the central limit theorem. At the end, we provide a discussion of the large deviations counterpart to our central limit theorem.


60D05 Geometric probability and stochastic geometry
60F05 Central limit and other weak theorems
60F10 Large deviations
60G15 Gaussian processes
Full Text: DOI arXiv Euclid


[1] Alonso-Gutiérrez, D., Prochno, J. and Thäle, C. (2018). Large deviations for high-dimensional random projections of \(\ell_p^n\) -balls. Adv. in Appl. Math.99 1-35. · Zbl 1391.60046
[2] Barthe, F., Guédon, O., Mendelson, S. and Naor, A. (2005). A probabilistic approach to the geometry of the \(l^n_p\) -ball. Ann. Probab.33 480-513. · Zbl 1071.60010
[3] Bass, R.F. (2011). Stochastic Processes. Cambridge Series in Statistical and Probabilistic Mathematics33. Cambridge: Cambridge Univ. Press.
[4] Basu, S.K. (1974). Density versions of the univariate central limit theorem. Ann. Probab.2 270-276. · Zbl 0304.60012 · doi:10.1214/aop/1176996708
[5] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability38. Berlin: Springer. Corrected reprint of the second (1998) edition. · Zbl 0896.60013
[6] den Hollander, F. (2000). Large Deviations. Fields Institute Monographs14. Providence, RI: Amer. Math. Soc. · Zbl 0949.60001
[7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II. 2nd ed. New York: Wiley. · Zbl 0219.60003
[8] Gantert, N., Kim, S.S. and Ramanan, K. (2017). Large deviations for random projections of \(\ell^p\) balls. Ann. Probab.45 4419-4476. · Zbl 1459.60067 · doi:10.1214/16-AOP1169
[9] Goldstein, L. and Shao, Q.-M. (2009). Berry-Esseen bounds for projections of coordinate symmetric random vectors. Electron. Commun. Probab.14 474-485. · Zbl 1189.60051 · doi:10.1214/ECP.v14-1502
[10] Kabluchko, Z., Litvak, A.E. and Zaporozhets, D. (2015). Mean width of regular polytopes and expected maxima of correlated Gaussian variables. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 442 75-96. · Zbl 1381.52010 · doi:10.1007/s10958-017-3492-3
[11] Kabluchko, Z., Prochno, J. and Thäle, C. (2017). High-dimensional limit theorems for random vectors in \(\ell_p^n\) -balls. Commun. Contemp. Math. To appear. · Zbl 1412.52007
[12] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). New York: Springer. · Zbl 0996.60001
[13] Klartag, B. (2007). A central limit theorem for convex sets. Invent. Math.168 91-131. · Zbl 1144.60021 · doi:10.1007/s00222-006-0028-8
[14] Klartag, B. (2007). Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal.245 284-310. · Zbl 1140.52004 · doi:10.1016/j.jfa.2006.12.005
[15] Klartag, B. (2009). A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields145 1-33. · Zbl 1171.60322 · doi:10.1007/s00440-008-0158-6
[16] Meckes, M.W. (2009). Gaussian marginals of convex bodies with symmetries. Beitr. Algebra Geom.50 101-118. · Zbl 1162.60006
[17] Naor, A. (2007). The surface measure and cone measure on the sphere of \(l_p^n\) . Trans. Amer. Math. Soc.359 1045-1079. · Zbl 1109.60006 · doi:10.1090/S0002-9947-06-03939-0
[18] Paouris, G., Pivovarov, P. and Zinn, J. (2014). A central limit theorem for projections of the cube. Probab. Theory Related Fields159 701-719. · Zbl 1301.52017 · doi:10.1007/s00440-013-0518-8
[19] Schechtman, G. and Zinn, J. (1990). On the volume of the intersection of two \(L^n_p\) balls. Proc. Amer. Math. Soc.110 217-224. · Zbl 0704.60017
[20] Simon, B. (2015). Basic Complex Analysis. A Comprehensive Course in Analysis, Part 2A. Providence, RI: Amer. Math. Soc. · Zbl 1332.00004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.