Concentration inequalities for polynomials in α-sub-exponential random variables

Friedrich Götze; Holger Sambale; Arthur Sinulis

doi:10.1214/21-EJP606

2021 Concentration inequalities for polynomials in α-sub-exponential random variables

Friedrich Götze, Holger Sambale, Arthur Sinulis

Author Affiliations +

Electron. J. Probab. 26: 1-22 (2021). DOI: 10.1214/21-EJP606

Abstract

We derive multi-level concentration inequalities for polynomials in independent random variables with an α-sub-exponential tail decay. A particularly interesting case is given by quadratic forms $f({X_{1}},\dots ,{X_{n}})=\langle X,AX\rangle$ , for which we prove Hanson–Wright-type inequalities with explicit dependence on various norms of the matrix A. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in α-sub-exponential random variables, such as quadratic Poisson chaos.

We provide various applications of these inequalities. Among them are generalizations of some results proven by Rudelson and Vershynin from sub-Gaussian to α-sub-exponential random variables, i. e. concentration of the Euclidean norm of the linear image of a random vector and concentration inequalities for the distance between a random vector and a fixed subspace.

References

1.

Radosław Adamczak, Moment inequalities for U-statistics, Ann. Probab. 34 (2006), no. 6, 2288–2314. MR2294982Radosław Adamczak, Moment inequalities for U-statistics, Ann. Probab. 34 (2006), no. 6, 2288–2314. MR2294982

2.

Radosław Adamczak, A note on the Hanson–Wright inequality for random vectors with dependencies, Electron. Commun. Probab. 20 (2015), no. 72, 13. MR3407216Radosław Adamczak, A note on the Hanson–Wright inequality for random vectors with dependencies, Electron. Commun. Probab. 20 (2015), no. 72, 13. MR3407216

3.

Radosław Adamczak and Rafał Latała, Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails, Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 4, 1103–1136. MR3052405Radosław Adamczak and Rafał Latała, Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails, Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 4, 1103–1136. MR3052405

4.

Radosław Adamczak and Paweł Wolff, Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order, Probab. Theory Related Fields 162 (2015), no. 3-4, 531–586. MR3383337Radosław Adamczak and Paweł Wolff, Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order, Probab. Theory Related Fields 162 (2015), no. 3-4, 531–586. MR3383337

5.

Sergey G. Bobkov, The growth of

{L^{p}}

-norms in presence of logarithmic Sobolev inequalities, Vestnik Syktyvkar Univ. 11 (2010), no. 2, 92–111.Sergey G. Bobkov, The growth of

{L^{p}}

-norms in presence of logarithmic Sobolev inequalities, Vestnik Syktyvkar Univ. 11 (2010), no. 2, 92–111.

6.

Sergey G. Bobkov, Friedrich Götze, and Holger Sambale, Higher order concentration of measure, Commun. Contemp. Math. 21 (2019), no. 03. MR3947067Sergey G. Bobkov, Friedrich Götze, and Holger Sambale, Higher order concentration of measure, Commun. Contemp. Math. 21 (2019), no. 03. MR3947067

7.

Sergey G. Bobkov and Michel Ledoux, Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution, Probab. Theory Related Fields 107 (1997), no. 3, 383–400. MR1440138Sergey G. Bobkov and Michel Ledoux, Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution, Probab. Theory Related Fields 107 (1997), no. 3, 383–400. MR1440138

8.

Víctor H. de la Peña and Evarist Giné, Decoupling, Probability and its Applications, Springer-Verlag, New York, 1999. MR1666908Víctor H. de la Peña and Evarist Giné, Decoupling, Probability and its Applications, Springer-Verlag, New York, 1999. MR1666908

9.

László Erdős, Horng-Tzer Yau, and Jun Yin, Bulk universality for generalized Wigner matrices, Probab. Theory Related Fields 154 (2012), no. 1-2, 341–407. MR2981427László Erdős, Horng-Tzer Yau, and Jun Yin, Bulk universality for generalized Wigner matrices, Probab. Theory Related Fields 154 (2012), no. 1-2, 341–407. MR2981427

10.

Efim D. Gluskin and Stanisław Kwapień, Tail and moment estimates for sums of independent random variables with logarithmically concave tails, Studia Math. 114 (1995), no. 3, 303–309. MR1338834Efim D. Gluskin and Stanisław Kwapień, Tail and moment estimates for sums of independent random variables with logarithmically concave tails, Studia Math. 114 (1995), no. 3, 303–309. MR1338834

11.

Friedrich Götze and Holger Sambale, Higher order concentration in presence of Poincaré-type inequalities, High Dimensional Probability VIII, Birkhäuser, Springer, Progress in Probability 74 (2019), pp. 55–69.Friedrich Götze and Holger Sambale, Higher order concentration in presence of Poincaré-type inequalities, High Dimensional Probability VIII, Birkhäuser, Springer, Progress in Probability 74 (2019), pp. 55–69.

12.

Friedrich Götze, Holger Sambale, and Arthur Sinulis, Concentration inequalities for bounded functionals via generalized log-Sobolev inequalities, J. Theor. Probab. (2020)Friedrich Götze, Holger Sambale, and Arthur Sinulis, Concentration inequalities for bounded functionals via generalized log-Sobolev inequalities, J. Theor. Probab. (2020)

13.

Friedrich Götze, Holger Sambale, and Arthur Sinulis, Higher order concentration for functions of weakly dependent random variables, Electron. J. Probab. 24 (2019), no. 85, 19 pp. MR4003138Friedrich Götze, Holger Sambale, and Arthur Sinulis, Higher order concentration for functions of weakly dependent random variables, Electron. J. Probab. 24 (2019), no. 85, 19 pp. MR4003138

14.

David L. Hanson and Farroll T. Wright, A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Statist. 42 (1971), no. 3, 1079–1083. MR0279864David L. Hanson and Farroll T. Wright, A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Statist. 42 (1971), no. 3, 1079–1083. MR0279864

15.

Paweł Hitczenko, Stephen J. Montgomery-Smith, and Krzysztof Oleszkiewicz, Moment inequalities for sums of certain independent symmetric random variables, Studia Math. 123 (1997), no. 1, 15–42. MR1438303Paweł Hitczenko, Stephen J. Montgomery-Smith, and Krzysztof Oleszkiewicz, Moment inequalities for sums of certain independent symmetric random variables, Studia Math. 123 (1997), no. 1, 15–42. MR1438303

16.

Christian Houdré and Patricia Reynaud-Bouret, Exponential inequalities, with constants, for U-statistics of order two, Stochastic inequalities and applications, Birkhäuser, Springer, Progress in Probability 56 (2003), pp. 55–69.Christian Houdré and Patricia Reynaud-Bouret, Exponential inequalities, with constants, for U-statistics of order two, Stochastic inequalities and applications, Birkhäuser, Springer, Progress in Probability 56 (2003), pp. 55–69.

17.

Daniel Hsu, Sham M. Kakade, and Tong Zhang, A tail inequality for quadratic forms of subgaussian random vectors, Electron. Commun. Probab. 17 (2012), no. 52, 6. MR2994877Daniel Hsu, Sham M. Kakade, and Tong Zhang, A tail inequality for quadratic forms of subgaussian random vectors, Electron. Commun. Probab. 17 (2012), no. 52, 6. MR2994877

18.

Konrad Kolesko and Rafał Latała, Moment estimates for chaoses generated by symmetric random variables with logarithmically convex tails, Statist. Probab. Lett. 107 (2015), 210–214. MR3412778Konrad Kolesko and Rafał Latała, Moment estimates for chaoses generated by symmetric random variables with logarithmically convex tails, Statist. Probab. Lett. 107 (2015), 210–214. MR3412778

19.

Stanisław Kwapień, Decoupling inequalities for polynomial chaos, Ann. Probab. 15 (1987), no. 3, 1062–1071. MR0893914Stanisław Kwapień, Decoupling inequalities for polynomial chaos, Ann. Probab. 15 (1987), no. 3, 1062–1071. MR0893914

20.

Rafał Latała, Tail and moment estimates for sums of independent random vectors with logarithmically concave tails, Studia Math. 118 (1996), no. 3, 301–304. MR1388035Rafał Latała, Tail and moment estimates for sums of independent random vectors with logarithmically concave tails, Studia Math. 118 (1996), no. 3, 301–304. MR1388035

21.

Rafał Latała, Tail and moment estimates for some types of chaos, Studia Math. 135 (1999), no. 1, 39–53. MR1686370Rafał Latała, Tail and moment estimates for some types of chaos, Studia Math. 135 (1999), no. 1, 39–53. MR1686370

22.

Rafał Latała, Estimates of moments and tails of Gaussian chaoses, Ann. Probab. 34 (2006), no. 6, 2315–2331. MR2294983Rafał Latała, Estimates of moments and tails of Gaussian chaoses, Ann. Probab. 34 (2006), no. 6, 2315–2331. MR2294983

23.

Rafał Latała and Rafał Łochowski, Moment and tail estimates for multidimensional chaos generated by positive random variables with logarithmically concave tails, Stochastic inequalities and applications, Progr. Probab., vol. 56, Birkhäuser, Basel, 2003, pp. 77–92. MR2073428Rafał Latała and Rafał Łochowski, Moment and tail estimates for multidimensional chaos generated by positive random variables with logarithmically concave tails, Stochastic inequalities and applications, Progr. Probab., vol. 56, Birkhäuser, Basel, 2003, pp. 77–92. MR2073428

24.

Michel Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. MR1849347Michel Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. MR1849347

25.

Rafał Meller, Two-sided moment estimates for a class of nonnegative chaoses, Statist. Probab. Lett. 119 (2016), 213–219. MR3555289Rafał Meller, Two-sided moment estimates for a class of nonnegative chaoses, Statist. Probab. Lett. 119 (2016), 213–219. MR3555289

26.

Rafał Meller, Tail and moment estimates for a class of random chaoses of order two, Studia Math. 249 (2019), no. 1, 1–32. MR3984282Rafał Meller, Tail and moment estimates for a class of random chaoses of order two, Studia Math. 249 (2019), no. 1, 1–32. MR3984282

27.

Alexey Naumov, Vladimir Spokoiny, and Vladimir Ulyanov, Bootstrap confidence sets for spectral projectors of sample covariance, Probab. Theory Related Fields 174 (2019), no. 3-4, 1091–1132. MR3980312Alexey Naumov, Vladimir Spokoiny, and Vladimir Ulyanov, Bootstrap confidence sets for spectral projectors of sample covariance, Probab. Theory Related Fields 174 (2019), no. 3-4, 1091–1132. MR3980312

28.

Joscha Prochno, Christoph Thäle, and Nicola Turchi, The isotropic constant of random polytopes with vertices on convex surfaces, J. Complexity 54 (2019), 101394, 17 pp. MR3983208Joscha Prochno, Christoph Thäle, and Nicola Turchi, The isotropic constant of random polytopes with vertices on convex surfaces, J. Complexity 54 (2019), 101394, 17 pp. MR3983208

29.

Mark Rudelson and Roman Vershynin, Hanson–Wright inequality and sub-Gaussian concentration, Electron. Commun. Probab. 18 (2013), no. 82, 9. MR3125258Mark Rudelson and Roman Vershynin, Hanson–Wright inequality and sub-Gaussian concentration, Electron. Commun. Probab. 18 (2013), no. 82, 9. MR3125258

30.

Holger Sambale, Some notes on concentration for α-subexponential random variables, arXiv preprint (2020).Holger Sambale, Some notes on concentration for α-subexponential random variables, arXiv preprint (2020).

31.

Holger Sambale and Arthur Sinulis, Logarithmic Sobolev inequalities for finite spin systems and applications, Bernoulli 26 (2020), no. 3, 1863–1890. MR4091094Holger Sambale and Arthur Sinulis, Logarithmic Sobolev inequalities for finite spin systems and applications, Bernoulli 26 (2020), no. 3, 1863–1890. MR4091094

32.

Warren Schudy and Maxim Sviridenko, Bernstein-like concentration and moment inequalities for polynomials of independent random variables: multilinear case, arXiv preprint (2011).Warren Schudy and Maxim Sviridenko, Bernstein-like concentration and moment inequalities for polynomials of independent random variables: multilinear case, arXiv preprint (2011).

33.

Warren Schudy and Maxim Sviridenko, Concentration and moment inequalities for polynomials of independent random variables, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2012, pp. 437–446. MR3205227Warren Schudy and Maxim Sviridenko, Concentration and moment inequalities for polynomials of independent random variables, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2012, pp. 437–446. MR3205227

34.

Van H. Vu and Ke Wang, Random weighted projections, random quadratic forms and random eigenvectors, Random Structures Algorithms 47 (2015), no. 4, 792–821. MR3418916Van H. Vu and Ke Wang, Random weighted projections, random quadratic forms and random eigenvectors, Random Structures Algorithms 47 (2015), no. 4, 792–821. MR3418916

35.

Farroll T. Wright, A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric, Ann. Probability 1 (1973), no. 6, 1068–1070. MR0353419Farroll T. Wright, A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric, Ann. Probability 1 (1973), no. 6, 1068–1070. MR0353419

36.

Zhuoran Yang, Lin F. Yang, Ethan X. Fang, Tuo Zhao, Zhaoran Wang, and Matey Neykov, Misspecified nonconvex statistical optimization for sparse phase retrieval, Math. Program. 176 (2019), no. 1-2, Ser. B, 545–571. MR3960819Zhuoran Yang, Lin F. Yang, Ethan X. Fang, Tuo Zhao, Zhaoran Wang, and Matey Neykov, Misspecified nonconvex statistical optimization for sparse phase retrieval, Math. Program. 176 (2019), no. 1-2, Ser. B, 545–571. MR3960819

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Friedrich Götze, Holger Sambale, and Arthur Sinulis "Concentration inequalities for polynomials in α-sub-exponential random variables," Electronic Journal of Probability 26(none), 1-22, (2021). https://doi.org/10.1214/21-EJP606

Received: 10 September 2019; Accepted: 12 March 2021; Published: 2021

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