Modified log-Sobolev inequalities and two-level concentration

Sambale H, Sinulis A (2021)
ALEA: Latin American Journal of Probability and Mathematical Statistics 18(1): 855-885.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form Ent(mu)(e(f)) <= rho/2 E-mu, e(f) Gamma(f )(2) for some difference operator Gamma, and show how it implies two-level concentration inequalities akin to the Hanson-Wright or Bernstein inequality. This can be applied to the continuous (e. g. the sphere or bounded perturbations of product measures) as well as discrete setting (the symmetric group, finite measures satisfying an approximate tensorization property, ... ). Moreover, we use modified logarithmic Sobolev inequalities on the symmetric group S-n and for slices of the hypercube to prove Talagrand's convex distance inequality, and provide concentration inequalities for locally Lipschitz functions on S-n. Some examples of known statistics are worked out, for which we obtain the correct order of fluctuations, which is consistent with central limit theorems.
Stichworte
Bernstein inequality; concentration of measure phenomenon; convex; distance inequality; Hanson-Wright inequality; modified logarithmic; Sobolev inequality; symmetric group
Erscheinungsjahr
2021
Zeitschriftentitel
ALEA: Latin American Journal of Probability and Mathematical Statistics
Band
18
Ausgabe
1
Seite(n)
855-885
eISSN
1980-0436
Page URI
https://pub.uni-bielefeld.de/record/2954697

Zitieren

Sambale H, Sinulis A. Modified log-Sobolev inequalities and two-level concentration. ALEA: Latin American Journal of Probability and Mathematical Statistics . 2021;18(1):855-885.
Sambale, H., & Sinulis, A. (2021). Modified log-Sobolev inequalities and two-level concentration. ALEA: Latin American Journal of Probability and Mathematical Statistics , 18(1), 855-885. https://doi.org/10.30757/ALEA.v18-31
Sambale, H., and Sinulis, A. (2021). Modified log-Sobolev inequalities and two-level concentration. ALEA: Latin American Journal of Probability and Mathematical Statistics 18, 855-885.
Sambale, H., & Sinulis, A., 2021. Modified log-Sobolev inequalities and two-level concentration. ALEA: Latin American Journal of Probability and Mathematical Statistics , 18(1), p 855-885.
H. Sambale and A. Sinulis, “Modified log-Sobolev inequalities and two-level concentration”, ALEA: Latin American Journal of Probability and Mathematical Statistics , vol. 18, 2021, pp. 855-885.
Sambale, H., Sinulis, A.: Modified log-Sobolev inequalities and two-level concentration. ALEA: Latin American Journal of Probability and Mathematical Statistics . 18, 855-885 (2021).
Sambale, Holger, and Sinulis, Arthur. “Modified log-Sobolev inequalities and two-level concentration”. ALEA: Latin American Journal of Probability and Mathematical Statistics 18.1 (2021): 855-885.

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