## Bernoulli

• Bernoulli
• Volume 8, Number 6 (2002), 697-720.

### Concentration and deviation inequalities in infinite dimensions via covariance representations

#### Abstract

Concentration and deviation inequalities are obtained for functionals on Wiener space, Poisson space or more generally for normal martingales and binomial processes. The method used here is based on covariance identities obtained via the chaotic representation property, and provides an alternative to the use of logarithmic Sobolev inequalities. It enables the recovery of known concentration and deviation inequalities on the Wiener and Poisson space (including those given by sharp logarithmic Sobolev inequalities), and extends results available in the discrete case, i.e. on the infinite cube $\{-1,1\}^\infty$.

#### Article information

Source
Bernoulli, Volume 8, Number 6 (2002), 697-720.

Dates
First available in Project Euclid: 9 February 2004

https://projecteuclid.org/euclid.bj/1076364802

Mathematical Reviews number (MathSciNet)
MR1962538

Zentralblatt MATH identifier
1012.60020

#### Citation

Houdré, Christian; Privault, Nicolas. Concentration and deviation inequalities in infinite dimensions via covariance representations. Bernoulli 8 (2002), no. 6, 697--720. https://projecteuclid.org/euclid.bj/1076364802

#### References

• [1] Ané, C. (2000) Grandes déviations et inégalités fonctionnelles pour des processus de Markov à temps continu sur un graphe. Doctoral thesis, Université Paul Sabatier - Toulouse III.
• [2] Ané, C. and Ledoux, M. (2000) On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields, 116, 573-602.
• [3] Bobkov, S.G. (1995) On the Gross and Talagrand inequalities on the discrete cube. Vestn. Syktyvkar. Univ. Ser. 1 Mat. Mekh. Inform., 1, 12-19.
• [4] Bobkov, S.G. and Ledoux, M. (1998) On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal., 156, 347-365. Abstract can also be found in the ISI/STMA publication
• [5] Bobkov, S.G., Götze, F. and Houdré, C. (2001a) On Gaussian and Bernoulli covariance representations. Bernoulli, 7, 439-451. Abstract can also be found in the ISI/STMA publication
• [6] Bobkov, S.G., Houdré, C. and Tetali, P. (2001b) The subgaussian constant and deviation inequalities. Preprint.
• [7] Borell, C. (1975) The Brunn-Minkowski inequality in Gauss space. Invent. Math., 30 207-216.
• [8] Carlen, E. and Pardoux, E. (1990) Differential calculus and integration by parts on Poisson space. In S. Albeverio, P. Blanchard and D. Testard (eds), Stochastics, Algebra and Analysis in Classical and Quantum Dynamics (Marseille, 1988), Math. Appl. 59, pp. 63-73. Dordrecht: Kluwer Academic.
• [9] Dieudonné, J. (1968) E´ léments d´analyse. Tome II: Chapitres XII à XV. Paris: Gauthier-Villars.
• [10] Elliott, R.J. and Tsoi, A.H. (1993) Integration by parts for Poisson processes. J. Multivariate Anal., 44, 197-190. Abstract can also be found in the ISI/STMA publication
• [11] E´ mery, M. (1989) On the Azéma martingales. In J. Azéma, P.A. Meyer and M. Yor (eds), Séminaire de Probabilités XXIII, Lecture Notes in Math. 1372, pp. 66-87. Berlin: Springer-Verlag.
• [12] Holden, H., Linstrøm, T., Øksendal, B. and Ubøe, J. (1992) Discrete Wick calculus and stochastic functional equations. Potential Anal., 1, 291-306.
• [13] Houdré, C. (2002) Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab., 30, 1223-1237.
• [14] Houdré, C. and Tetali, P. (2001) Concentration of measure for products of Markov kernels and graph products via functional inequalities. Combin. Probab. Comput., 10, 1-28.
• [15] Ito, Y. (1988) Generalized Poisson functionals. Probab. Theory Related Fields, 77, 1-28.
• [16] Ledoux, M. (1996a) On Talagrand´s deviation inequalities for product measures. ESAIM Probab. Statist., 1, 63-87.
• [17] Ledoux, M. (1996b) Isoperimetry and Gaussian analysis. In P. Bernard (ed.), Lectures on Probability Theory and Statistics: E´ cole d´E´ té de Probabilités de Saint-Flour XXIV - 1994, Lecture Notes in Math. 1648, pp. 165-294. Berlin: Springer-Verlag.
• [18] Ledoux, M. (1999) Concentration of measure and logarithmic Sobolev inequalities. In J. Azéma, M. E´ mery, M. Ledoux and M. Yor (eds), Séminaire de Probabilités XXXIII, Lecture notes in Math. 1709, pp. 120-216. Berlin: Springer-Verlag.
• [19] Ledoux, M. (2000) The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6), 9, 305-366.
• [20] Leitz-Martini, M. (2000) A discrete Clark-Ocone formula. MaPhySto Research Report No. 29.
• [21] Nualart, D. and Vives, J. (1995) A duality formula on the Poisson space and some applications. In Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), pp. 205-213. Basel: Birkhäuser.
• [22] Pisier, G. (1986) Probabilistic methods in the geometry of Banach spaces. In G. Letta and M. Pratelli (eds), Probability and Analysis, Lecture Notes in Math. 1206, pp. 167-241. Berlin: Springer- Verlag.
• [23] Privault, N. (1994a) Chaotic and variational calculus in discrete and continuous time for the Poisson process. Stochastics Stochastics Rep., 51, 83-109. Abstract can also be found in the ISI/STMA publication
• [24] Privault, N. (1994b) Inégalités de Meyer sur l´espace de Poisson. C. R. Acad. Sci. Paris Sér. I. Math., 318, 559-562.
• [25] Privault, N. (1999) Independence of a class of multiple stochastic integrals. In R. Dalang, M. Dozzi and F. Russo (eds), Seminar on Stochastic Analysis, Random Fields and Applications (Ascona,
• [26] 1996), Progr. Probab. 45, pp. 249-259. Basel: Birkhäuser.
• [27] Privault, N. and Schoutens, W. (2002) Discrete chaotic calculus covariance identities. Stochastics Stochastics Rep., 72, 289-315.
• [28] Reynaud-Bouret, P. (2001) Concentration inequalities for inhomogeneous Poisson processes and adaptative estimation of the density. Prépublication d´Orsay No. 2001-18.
• [29] Sudakov, V.N. and Tsirel´son, B.S. (1974) Extremal properties of half-spaces for spherically invariant measures. Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. Steklov., 41, 14-24.
• [30] Surgailis, D. (1984) On multiple Poisson stochastic integrals and associated Markov semi-groups. Probab. Math. Statist., 3, 217-239.