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Index

Adaptiveestimation, 85, 102, 139, 144, 219,

251, 267, 303estimator, 98, 102, 142, 203, 204, 270,

271, 303, 305Akaike, 7, 89, 226, 227, 236Akaike’s criterion, 7, 225–227, 236Aldous, 108Alexander, 139Approximation theory, 80, 88, 111, 251,

255, 266

Baraud, 181Barron, 274Bartlett, 287Bayes, 2, 106, 279, 284, 296Besov

ball, 99, 115, 121, 144body, 107, 121, 122, 125, 144, 146,

263, 266ellipsoid, 111, 115, 132, 133, 134,

136, 138, 139, 143semi-norm, 144

Bias term, 6, 89, 91, 92, 102, 116, 120,124, 134, 244, 245, 266, 267, 269,270, 293

Birge, 32, 143, 144Birge’s lemma, 32, 34, 102, 109, 253BirgeE, 31Borel–Cantelli lemma, 54, 82Borell’s theorem, 60Boucheron, 287, 317Bounded regression, 279, 293, 303, 308,

311

Bousquet, 170, 208, 288Brownian

motion, 61, 79, 83sheet, 3, 5, 84

Central Limit Theorem, 1, 2, 64Chaining argument, 17, 72, 183, 186,

189, 190Change points detection, 8, 91, 94, 316Chernoff, 15, 21, 159Chi-square statistics, 170, 171, 172,

207–209, 211, 227, 228, 230, 231,234, 237, 251

Classification, 2, 5, 279, 280, 283,284, 287, 293, 296, 302, 303, 308,311–313, 316

Classifier, 2, 5, 186, 279, 284, 286, 296,316

Combinatorial entropy, 157, 160, 187,188, 284, 286, 297, 298

Concentration Inequalities, 9Concentration inequalities, 2, 8, 9, 12,

26, 27, 39, 40, 58, 60, 65, 77, 147,157, 167, 170, 171, 227, 251, 280,317

Coupling, 35–37, 39, 51, 168Cramer, 15, 21, 159Cramer transform, 16, 19, 32, 34, 108Cross-validation, 203, 204, 205, 215

Data-driven penalty, 157, 208, 215, 314,316

Density estimation, 2, 4, 7, 201, 204,205, 251, 257, 303, 305, 306

326 Index

Donoho, 7, 85, 89, 92, 99, 102, 205,267

Donsker, 183, 297Duality formula

for entropy, 12, 28, 29–32, 40, 41, 48,166

for φ-entropy, 44, 46, 48, 49Dudley, 1, 70, 74, 81Dudley’s criterion, 71, 74, 75, 76, 80

Efron, 50Ehrhardt, 60Ellipsoid, 80, 82, 85, 86, 107, 113,

115–117, 119, 121, 122, 131, 132,135–137, 219, 254, 255, 257, 263,264, 266, 267

Empirical coefficients, 93, 204, 205Empirical contrast, 3, 3, 7, 202Empirical criterion, 3, 3, 4, 6Empirical measure, 171, 183, 203,

208, 209, 305, 308, 310,313

Empirical process, 1, 9, 9, 11, 12, 17,19, 74, 148, 151, 153, 157, 162,167, 168, 170, 180, 183–186, 192,206, 227, 281, 287–289, 297, 301,312

Empirical projection, 88, 203Empirical risk, 280, 289, 290, 301, 312

minimization, 5, 279, 280, 311minimizer, 280, 280, 287–290, 299,

301, 312Entropy, 12, 27, 29, 30, 40, 43, 46, 48,

62, 154, 157Entropy method, 12, 148, 154, 170Entropy with bracketing, 183, 184, 190,

238, 239, 245, 247, 251, 257, 271,273, 294, 297, 300

Fernique, 56, 70, 76

Gaussian process, 53, 70, 72, 74, 77, 78,183

Gaussian regression, 83, 86, 87Gaussian sequence, 85, 87, 99, 115, 122,

125, 132, 136Girsanov’s formula, 102Gross, 62

Holder class, 252, 268, 305, 307Holder smooth, 79, 80, 273, 296, 304Hamming distance, 10, 11, 105, 107,

147, 151Haussler’s bound, 187, 189Herbst argument, 31, 62Hilbert space, 78, 79, 83, 87, 115Hilbert–Schmidt ellipsoid, 76, 80, 131Histogram, 171, 172, 201, 204, 225, 225,

232, 236–238, 250, 296Hoeffding, 21, 36, 40, 148, 153, 162Hoeffding’s lemma, 21, 31Hold-out, 307, 308, 310, 312, 313Hypothesis testing, 31

Ideal model, 6, 204Inequalities

Bennett, 23, 26, 30, 34Bernstein, 22, 24, 26, 30, 36, 148,

168, 170, 185, 192, 211, 217, 222,235, 243, 310

Bienayme–Chebycheff, 54Borell, 56Bousquet, 170, 291Burkholder, 172Cauchy–Schwarz, 37, 40, 47, 65, 81,

130, 149, 171, 188, 196, 207, 209,

210, 213, 224, 228, 230, 235

Chernoff, 16, 18–20, 22, 24–26, 62,105, 108, 149, 159

Cirelson–Ibragimov–Sudakov, 10, 56,61

Efron–Stein, 12, 13, 51, 51, 155, 167,168, 173, 176

Gross, 62Holder, 16, 173, 177Han, 42, 158, 160, 161Hoeffding, 21, 22, 22, 26Jensen, 17, 27, 38, 43, 46, 47, 108,

130, 188, 209, 264, 284–286Logarithmic Sobolev, 30, 56, 61, 62,

165Markov, 15, 18, 75, 277Marton, 37, 148, 149maximal, 139, 183–187, 190, 241, 276,

297Mc Diarmid, 148, 281moment, 155, 166, 174, 314, 317Pinsker, 31, 32, 35–38

Index 327

Sobolev, 155, 162

sub-Gaussian, 10, 11, 188

symmetrization

for empirical processes, 170, 188,189

for phi-entropy, 156, 162

Talagrand, 9, 11, 13, 169, 170, 180,208, 288

tensorization for entropy, 27, 40, 42,51, 62, 63, 157

tensorization for φ-entropy, 43, 44,49, 154, 155, 162–165

tensorization for the variance, 50,155

Isonormal process, 56, 76, 77, 79–81,83, 84, 103, 115, 125, 131

Johnstone, 7, 85, 89, 92, 99, 102, 205,267

Kerkyacharian, 102, 205, 267

Koltchinskii, 287

Kullback–Leibler

bias, 236, 267

information, 4, 27, 102, 104, 158,201, 254, 257, 273–275, 313

loss, 202, 225–227, 231, 236, 244, 245,253, 256, 257

projection, 225

risk, 225, 232, 233, 236

Levy, 58, 61

Levy’s isoperimetric theorem, 58

Latala, 44, 46, 154

Latala–Oleszkiewicz condition, 43

Least squares criterion

in the classification framework, 5, 284

in the density framework, 4, 202, 203

in the Gaussian framework, 5, 87

in the regression framework, 4, 283

Least squares estimator (LSE)

in the density framework, 202, 203,204, 212

in the Gaussian framework, 6, 87, 88,89, 106, 126, 136, 141

in the regression framework, 293,296, 305

Lebarbier, 316

Ledoux, 12, 18, 40, 44, 46, 58, 153, 157,168

Lepskii’s method, 102

Linear model, 83, 88, 89, 91, 99, 131,201, 204, 316

Lipschitz function, 55–62, 147, 150

Loss function, 3–5, 9, 87, 202, 225, 281,284, 287, 293, 303, 314

�p-body, 85, 86, 105–107, 115, 116, 119,121, 144, 146

Lugosi, 287

Mallows, 7, 89

Mallows’ Cp, 7, 89, 97, 98, 316

Margin condition, 296, 300

Marton, 12, 36, 148

Maximum likelihood

criterion, 4, 104, 201, 204, 226, 313

estimator (MLE), 1, 24, 143, 144,225, 240, 257, 267, 270, 308

Metric entropy, 56, 70, 71, 74–76, 80,110, 111, 139, 141, 183, 257, 260,273, 294

Minimax

estimator, 99, 102, 106, 110, 117,119, 122, 125, 134, 138, 142, 143,251, 269, 271, 273, 307

lower bound, 12, 27, 31, 105, 106, 110,111, 117, 122, 134, 252, 257, 260,263, 269, 296

point of view, 101, 296, 300

risk, 32, 101, 102, 106, 107, 122, 142,144, 237, 263, 265, 267, 271, 272,293, 305

Minimum contrast

estimation, 3, 5, 279, 280

estimator, 3, 6, 7, 316

Model selection, 1, 2, 8, 11, 56, 283,301, 303, 308, 313

in the classification framework, 157,284, 302, 303

in the density framework, 201, 202,204, 206–208, 211, 212, 238,251

328 Index

in the Gaussian framework, 86, 88,89, 98, 102, 115, 122, 125, 131,201, 316

in the regression framework, 181, 303,305

Moment generating function, 15, 17,23, 27, 29, 30, 32, 51, 56, 61, 108,148, 157, 168, 169

Net, 71, 76, 110, 112, 114, 142–144,261, 262, 272, 301, 306, 307

Oleszkiewicz, 44, 46, 154Oracle, 7, 9, 89, 91, 92, 225, 226, 236,

237inequality, 92, 101, 116, 133, 236, 244,

312, 314, 316

Packing number, 71, 111Penalized criterion, 7, 8, 11, 201, 303,

316Penalized estimator, 8, 115, 203, 280,

281, 283, 285, 302, 303Penalized least squares criterion

in the density framework, 201, 203in the Gaussian framework, 90, 94,

100, 125, 136in the regression framework, 7, 303

Penalized log-likelihood criterion, 7,201, 227, 232, 237, 238, 240, 313

Penalized LSEin the density framework, 202,

203–206, 212, 216, 219, 220, 251,263, 265

in the Gaussian framework, 90,92–94, 98, 100–102, 110, 116, 118,120, 125, 126, 133, 134, 136, 142,143, 204

in the regression framework, 303,303, 304, 305, 307

Penalized MLE, 204, 227, 230, 232,235–237, 250, 267, 268, 270–273

Penalty function, 7, 7, 8, 90, 95, 99, 116,123, 136, 202–208, 212, 215–223,225, 230, 232, 235–237, 263, 265,268, 280, 283, 285, 286

φ-entropy, 43, 46, 48–50, 154, 155, 162Picard, 102, 205, 267

Piecewise polynomial, 8, 91, 122, 145,225, 238, 246, 247, 250, 268, 273,296, 305

Pinelis, 181Pinsker, 85Pisier, 17Product measure, 56, 59, 147Product space, 35–37, 39Prohorov distance, 36

Quadratic risk, 6, 88, 88, 89, 98, 106,221, 224, 293

Random variablesBernoulli, 20, 32, 34, 44, 62, 63, 160binomial, 20, 105, 107, 108, 160Gaussian, 18, 19, 53, 55, 69hypergeometric, 107, 108Poisson, 19, 23, 24, 157, 159, 160Rademacher, 22, 39, 63, 161, 184, 185

Regression function, 2, 2, 3, 5, 279, 284,293, 303, 305–307

Rio, 26, 170Risk bound, 91, 94, 98, 100, 126, 130,

133, 136, 142, 204, 206, 208, 220,227, 230, 235–237, 263, 265, 271,283, 285–287, 289, 299–301

Sauer’s lemma, 187, 300Schmidt, 58Self-bounding condition, 157, 160, 161Shannon entropy, 41Sheu, 274Slepian’s lemma, 55, 66, 69Sobolev ball, 85, 99, 102Statistical learning, 2, 161, 280, 286,

310, 316Stein, 50Sudakov’s minoration, 56, 69, 80

Talagrand, 2, 18, 56, 59, 76, 147, 148,150, 151, 168

Thresholding estimator, 92, 98, 99, 205,206, 265, 267

Total variation distance, 31, 35Transportation

cost, 30, 31, 36, 36, 37, 40, 148–150method, 36, 39, 149

Tsybakov, 296, 297, 300, 301

Universal entropy, 183, 187, 188

Index 329

Van der Vaart, 1Vapnik, 5, 279, 280, 283Variable selection, 86, 87, 91, 92,

94, 97–99, 107, 115,116

Variance term, 6, 89Varshamov–Gilbert’s lemma, 105, 106,

107

VC-class, 186, 285, 299, 300VC-dimension, 186, 285, 297, 298, 313

Wavelet, 8, 86, 91, 99, 102, 122, 144,145, 205, 211, 247, 255, 265

Wellner, 1White noise, 3, 5, 6, 79, 80, 83, 84, 86,

92, 102, 104, 143, 303

List of Participants

Lecturers

DEMBO Amir Stanford Univ., USAFUNAKI Tadahisa Univ. Tokyo, JapanMASSART Pascal Univ. Paris-Sud, Orsay, F

Participants

AILLOT Pierre Univ. Rennes 1, FATTOUCH Mohammed Kadi Univ. Djillali Liabes, Sidi Bel Abbes, Algerie

AUDIBERT Jean-Yves Univ. Pierre et Marie Curie, Paris, FBAHADORAN Christophe Univ. Blaise Pascal, Clermont-Ferrand, FBEDNORZ Witold Warsaw Univ., PolandBELARBI Faiza Univ. Djillali Liabes, Sidi Bel Abbes, Algerie

BEN AROUS Gerard Courant Institute, New York, USABLACHE Fabrice Univ. Blaise Pascal, Clermont-Ferrand, FBLANCHARD Gilles Univ. Paris-Sud, Orsay, FBOIVIN Daniel Univ. Brest, FCHAFAI Djalil Ecole Veterinaire Toulouse, FCHOUAF Benamar Univ. Djillali Liabes, Sidi Bel Abbes, Algerie

DACHIAN Serguei Univ. Blaise Pascal, Clermont-Ferrand, FDELMOTTE Thierry Univ. Paul Sabatier, Toulouse, FDJELLOUT Hacene Univ. Blaise Pascal, Clermont-Ferrand, FDUROT Cecile Univ. Paris-Sud, Orsay, F

332 List of Participants

FLORESCU Ionut Purdue Univ., West Lafayette, USAFONTBONA Joaquin Univ. Pierre et Marie Curie, Paris, FFOUGERES Pierre Imperial College, London, UKFROMONT Magalie Univ. Paris-Sud, Orsay, FGAIFFAS Stephane Univ. Pierre et Marie Curie, Paris, FGUERIBALLAH Abdelkader Univ. Djillali Liabes, Sidi Bel Abbes, Algerie

GIACOMIN Giambattista Univ. Paris 7, FGOLDSCHMIDT Christina Univ. Cambridge, UKGUSTO Gaelle INRA, Jouy en Josas, FHARIYA Yuu Kyoto Univ., JapanJIEN Yu-Juan Purdue Univ., West Lafayette, USAJOULIN Alderic Univ. La Rochelle, FKLEIN Thierry Univ. Versailles, FKLUTCHNIKOFF Nicolas Univ. Provence, Marseille, FLEBARBIER Emilie INRIA Rhone-Alpes, Saint-Ismier, FLEVY-LEDUC Celine Univ. Paris-Sud, Orsay, FMAIDA Mylene ENS Lyon, FMALRIEU Florent Univ. Rennes 1, FMARTIN James CNRS, Univ. Paris 7, FMARTY Renaud Univ. Paul Sabatier, Toulouse, FMEREDITH Mark Univ. Oxford, UKMERLE Mathieu ENS Paris, FMOCIOALCA Oana Purdue Univ., West Lafayette, USANISHIKAWA Takao Univ. Tokyo, JapanOBLOJ Jan Univ. Pierre et Marie Curie, Paris, FOSEKOWSKI Adam Warsaw Univ., PolandPAROUX Katy Univ. Besancon, FPASCU Mihai Purdue Univ., West Lafayette, USAPICARD Jean Univ. Blaise Pascal, Clermont-Ferrand, FREYNAUD-BOURET Patricia Georgia Instit. Technology, Atlanta, USARIOS Ricardo Univ. Central, Caracas, VenezuelaROBERTO Cyril Univ. Marne-la-Vallee, FROITERSHTEIN Alexander Technion, Haifa, Israel

List of Participants 333

ROUX Daniel Univ. Blaise Pascal, Clermont-Ferrand, FROZENHOLC Yves Univ. Paris 7, FSAINT-LOUBERT BIE Erwan Univ. Blaise Pascal, Clermont-Ferrand, FSCHAEFER Christin Fraunhofer Institut FIRST, Berlin, DSTOLTZ Gilles Univ. Paris-Sud, Orsay, FTOUZILLIER Brice Univ. Pierre et Marie Curie, Paris, FTURNER Amanda Univ. Cambridge, UKVERT Regis Univ. Paris-Sud, Orsay, FVIENS Frederi Purdue Univ., West Lafayette, USAVIGON Vincent INSA Rouen, FYOR Marc Univ. Pierre et Marie Curie, Paris, FZACHARUK Mariusz Univ. Wroc�law, PolandZEITOUNI Ofer Univ. Minnesota, Minneapolis, USAZHANG Tao Purdue Univ., West Lafayette, USAZWALD Laurent Univ. Paris-Sud, Orsay, F

List of Short Lectures

Jean-Yves AUDIBERT Aggregated estimators and empiricalcomplexity for least squares regression

Christophe BAHADORAN Convergence and local equilibrium forthe one-dimensional asymmetricexclusion process

Fabrice BLACHE Backward stochastic differentialequations on manifolds

Gilles BLANCHARD Some applications of model selection tostatistical learning procedures

Serguei DACHIAN Description of specifications by means ofprobability distributions in smallvolumes under condition of very weakpositivity

Thierry DELMOTTE How to use the stationarity of areversible random environment toestimate derivatives of the annealeddiffusion

Ionut FLORESCU Pricing the implied volatility surfaceJoaquin FONTBONA Probabilistic interpretation and

stochastic particle approximations of the3-dimensional Navier-Stokes equation

Pierre FOUGERES Curvature-dimension inequality andprojections; some applications toSobolev inequalities

336 List of Short Lectures

Magalie FROMONT Tests adaptatifs d’adequation dans unmodele de densite

Giambattista GIACOMIN On random co-polymers and disorderedwetting models

Christina GOLDSCHMIDT Critical random hypergraphs: astochastic process approach

Yuu HARIYA Large time limiting laws for Brownianmotion perturbed by normalizedexponential weights (Part II)

Alderic JOULIN Isoperimetric and functional inequalitiesin discrete settings: application to thegeometric distribution

Thierry KLEIN Processus empirique et concentrationautour de la moyenne

Celine LEVY-LEDUC Estimation de periodes de fonctionsperiodiques bruitees et de formeinconnue; applications a la vibrometrielaser

James MARTIN Particle systems, queues, and geodesicsin percolation

Renaud MARTY Theoreme limite pour une equationdifferentielle a coefficient aleatoire amemoire longue

Oana MOCIOALCA Additive summable processes and theirstochastic integral

Takao NISHIKAWA Dynamic entropic repulsion for theGinzburg-Landau ∇φ interface model

Jan OBLOJ The Skorokhod embedding problem forfunctionals of Brownian excursions

Katy PAROUX Convergence locale d’un modele booleende couronnes

Mihai N. PASCU Maximum principles forNeumann/mixed Dirichlet-Neumanneigenvalue problem

Patricia REYNAUD-BOURET Adaptive estimation of the Aalenmultiplicative intensity by modelselection

List of Short Lectures 337

Cyril ROBERTO Sobolev inequalities for probabilitymeasures on the real line

Alexander ROITERSHTEIN Limit theorems for one-dimensionaltransient random walks in Markovenvironments

Gilles STOLTZ Internal regret in on-line prediction ofindividual sequences and in on-lineportfolio selection

Amanda TURNER Convergence of Markov processes nearhyperbolic fixed points

Vincent VIGON Les abrupts et les erodesMarc YOR Large time limiting laws for Brownian

motion perturbed by normalizedexponential weights (Part I)

Ofer ZEITOUNI Recursions and tightness

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