Table Of ContentProgress in Probability
Volume 66
Series Editors
Davar Khoshnevisan
Sidney I. Resnick
For further volumes:
http://www.springer.com/series/4839
High Dimensional
Probability VI
The Banff Volume
Christian Houdré
David M. Mason
Jan Rosi(cid:276)ski
Jon A. Wellner
Editors
Editors
Christian Houdré (cid:68)(cid:97)(cid:118)(cid:105)(cid:100)(cid:32)(cid:77)(cid:46)(cid:32)(cid:77)(cid:97)(cid:115)(cid:111)(cid:110)
Georgia Institute of Technology University of Delaware
Atlanta (cid:78)(cid:101)(cid:119)(cid:97)(cid:114)(cid:107)
USA (cid:85)(cid:83)(cid:65)
Jan Rosiński Jon A. Wellner
University of Tennessee University of Washington
Knoxville Seattle
USA USA
ISBN 978-3-0348-0489-9 ISBN 978-3-0348-0490-5 (eBook)
DOI10.1007/978-3-0348-0490-5
Springer Basel Heidelberg New York Dordrecht London
Library of Congress Control Number: 2013936649
Mathematical Subject Classification (2010): 60-XX, 62-XX
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Contents
Preface .................................................................. vii
Participants .............................................................. xii
Dedication ............................................................... xiii
Part I: Inequalities and Convexity
F. Gao
Bracketing Entropy of High Dimensional Distributions .............. 3
J. Hoffmann-Jørgensen
Slepian’s Inequality, Modularity and Integral Orderings ............. 19
P. Kevei and D.M. Mason
A More General Maximal Bernstein-type Inequality ................. 55
R. Latal̷a
Maximal Inequalities for Centered Norms of Sums of
Independent Random Vectors ....................................... 63
M. Lifshits, R.L. Schilling and I. Tyurin
A Probabilistic Inequality Related to Negative Definite Functions ... 73
I. Pinelis
Optimal Re-centering Bounds, with Applications to
Rosenthal-type Concentration of Measure Inequalities ............... 81
J.A. Wellner
Strong Log-concavityis Preservedby Convolution ................... 95
P. Wolff
On Some Gaussian Concentration Inequality for
Non-Lipschitz Functions ............................................ 103
Part II: Limit Theorems
J. Dedecker, F. Merlev`ede and F. P`ene
Rates of Convergence in the Strong Invariance Principle
for Non-adapted Sequences. Application to Ergodic
Automorphisms of the Torus ........................................ 113
vi Contents
F. G¨otze and A. Tikhomirov
On the Rate of Convergence to the Semi-circular Law ............... 139
J. Kuelbs and J. Zinn
Empirical Quantile CLTs for Time-dependent Data ................. 167
M. Peligrad
Asymptotic Properties for Linear Processes of Functionals
of Reversible or Normal Markov Chains ............................. 195
Part III: Stochastic Processes
F. Aurzada and T. Kramm
First Exit of Brownian Motion from a One-sided
Moving Boundary ................................................... 213
A. Basse-O’Connor and J. Rosin´ski
On L´evy’s Equivalence Theorem in Skorohod Space ................. 219
M.B. Marcus and J. Rosen
Continuity Conditions for a Class of Second-order
Permanental Chaoses ............................................... 227
Part IV: Random Matrices and Applications
R. Adamczak
On the Operator Norm of Random Rectangular
Toeplitz Matrices ................................................... 247
H. D¨oring and P. Eichelsbacher
Edge Fluctuations of Eigenvalues of Wigner Matrices ............... 261
C. Houdr´e and H. Xu
On the Limiting Shape of Young Diagrams Associated
with Inhomogeneous Random Words ................................ 277
Part V: High Dimensional Statistics
V. Koltchinskii and P. Rangel
Low Rank Estimation of Similarities on Graphs ..................... 305
K. Lounici
Sparse Principal Component Analysis with Missing
Observations ........................................................ 327
D. Radulovic
High Dimensional CLT and its Applications ......................... 357
Preface
The High Dimensional Probability assemblage of probabilists grew out of a group
of mathematicians, who had a common interest in doing probability on Banach
spaces.TherewerenineInternationalConferences inProbability inBanach Spaces
beginning with Oberwolfachin 1975. An earlier conference on Gaussian processes
with many of the same participants as the 1975 meeting was held in Strasbourg,
Francein1973.The lastBanachspacemeeting tookplaceinSandjberg,Denmark
in 1993. It was decided in 1994, that in order to reflect the widening interests of
the members of the group, to change the name of this conference series to the
International Conference on High Dimensional Probability.
ThepresentvolumeisanoutgrowthoftheSixthHighDimensionalProbability
Conference (HDP VI) held at the Banff International Research Station (BIRS),
Banff, Canada, October 9–14, 2011. The scope and the quality of the contributed
papersshowverywell thathighdimensionalprobability(HDP) remainsa vibrant
and expanding areaof mathematical research.Four of the participants of the first
Probability on Banach Spaces meeting-Jørgen Hoffmann-Jørgensen, Jim Kuelbs,
Mike Marcus and Jan Rosin´ski-have contributed papers to this volume.
HDP deals with a set of ideas and techniques whose origin can largely be
traced back to the theory of Gaussian processes and, in particular, the study
of their paths properties. The original impetus was to characterize boundedness
or continuity via geometric structures associated with random variables in high
dimensional or infinite dimensional spaces. More precisely, these are geometric
characteristics of the parameter space, equipped with the metric induced by the
covariancestructure ofthe process,describedvia metric entropy,majorizingmea-
sures and generic chaining.
This original set of ideas and techniques turned out to be particularly fruit-
ful in extending the classical limit theorems in probability, such as laws of large
numbers, laws of iterated logarithm and central limit theorems, to the context of
Banach spaces and in the study of empirical processes.
Similardevelopmentstookplaceinothermathematicalsubfieldssuchascon-
vex geometry, asymptotic geometric analysis, additive combinatorics and random
matrices,tonamebutafewtopics.Moreover,themethodsofHDP,andespecially
itsoffshoot,theconcentrationofmeasurephenomenon,werefoundtohaveanum-
ber of important applications in these areasas well as in Statistics and Computer
viii Preface
Science. This breadth is very well illustrated by the contributions in the present
volume.
MostofthepapersinthisvolumewerepresentedatHDPVI.Theparticipants
of this conference are grateful for the support of the BIRS and the editors thank
Springer Verlag for agreeing to publish the resulting HDP VI volume.
The papers in this volume aptly display the methods and breadth of HDP.
They use a variety of techniques in their analysis that should be of interest to
advancedstudentsandresearchers.Wehaveorganizedthepapersintofivegeneral
areas:InequalitiesandConvexity,LimitTheorems,StochasticProcesses,Random
Matrices and High Dimensional Statistics. To give an idea of their scope, we shall
now briefly describe them by subject area.
Inequalities and Convexity:
∙ Bracketing entropy of high dimensional distributions, by Fuchang Gao
∙ Slepian’sinequality,modularityandintegralorderings,byJørgenHoffmann-
Jørgensen
∙ AmoregeneralmaximalBernstein-typeinequality,byP´eterKeveiandDavid
M. Mason
∙ Maximalinequalitiesforcenterednormsofsumsofindependentrandomvec-
tors, by Rafal̷ Lata̷la
∙ A probabilistic inequality related to negative definite functions, by Mikhail
Lifshits, Ren´e L. Schilling and Ilya Tyurin
∙ Optimal re-centering bound, with applications to Rosenthal-type concentra-
tion of measure inequalities, by Iosif Pinelis
∙ Strong log-concavityis preserved by convolution, by Jon A. Wellner
∙ On some Gaussian concentration inequality for non-Lipschitz functions, by
Pawe̷l Wolff
Gao considers the family of all distribution functions on [0,1]𝑑 and obtains
boundsonthebracketingentropiesoftheseclassesfor𝐿 metricswith𝑝≥1.These
𝑝
bounds have important implications for rates of convergence of nonparametric
estimators in a number of statistical problems.
Jørgen Hoffmann-Jørgensen proves a unified version of Slepian’s inequality
underminimalregularityconditionsandpointstothesubtletiesoftheassumptions
of this inequality. This basically covers all forms of Slepian’s inequality known in
the literature. Then the author explores the connection of Slepian’s inequality to
integral orderingsin general, and to the supermodular orderingin particular, and
also corrects some results in the theory of integral orderings.
Kevei and Mason show that under very weak assumptions a general version
of Bernstein’s exponential inequality for sums of random variables, which are not
necessarily independent, extends to a maximal version.
Lata̷laprovesa L´evy-Ottavianitype inequality for sums ofindependent ran-
domvariableswitharbitrarycentering.Thevectorcaseisexploredandamodified
inequality proved in the Hilbert space framework.
Preface ix
Lifshits,SchillingandTyurinproveaninequalitycomparingtheexpectations
ofanegativedefinitefunctionappliedtoeitherthedifferenceorthesumoftwoiid
random vectors. A particular case which involves lower powers of the Euclidean
norm is linked to bifractional Brownian motion.
Pinelisconsidersoptimalboundsbetweenexpectationsofanonnegativefunc-
tion of centered and non-centered random variables. Applications to Rosenthal-
type concentration of measure inequalities are given.
Although it is well known that log-concavity of distributions is preserved
underconvolution(additionofindependentrandomvariables),preservationofthe
stronger notion of ultra log-concavity under convolution in the setting of integer-
valuedrandomvariableswasfirstprovedbyLiggett(1997).Wellner’spapershows
that a recentproof of Liggett’s result by Johnson(2007)carries overto a proof of
the preservationof stronglog-concavityunder convolutionfor real-valuedrandom
variables.
Wolff’s paper establishes a concentration inequality for functions of a pair
of Gaussian random vectors. The bounded Lipschitz assumption present in the
classicalGaussianconcentrationinequalityis now replacedby the boundedness of
second-order derivatives.
Limit Theorems:
∙ Rates of convergence in the strong invariance principle for non adapted
sequences: application to ergodic automorphisms of the torus, by J´erˆome
Dedecker, Florence Merlev`ede and Fran¸coise P`ene
∙ On the rate of convergence to the semi-circular law, by Friedrich G¨otze and
Alexandre Tikhomirov
∙ EmpiricalquantileCLTsfortimedependentdata,byJamesKuelbsandJoel
Zinn
∙ Asymptotic propertiesfor linearprocessesoffunctionals ofreversibleornor-
mal Markov chains, by Magda Peligrad
Dedecker,Merlev`edeandP`eneestablishstronginvarianceprinciplesfornon-
adapted sequences and apply them to iterates of ergodic automorphisms of the
𝑑-dimensionaltorus.Theirmaintheoremsareprovedusinganapproximatingmar-
tingale introduced by Gordin (1969).
Applying a bound for the Kolmogorov distance between distribution func-
tions via Stieltjes transforms, G¨otze and Tikhomirov derive under side conditions
a rate of convergence in the semi-circular law.
Peligrad proves central limit theorems for linear processes of functionals of
reversible or normal Markov chains. The proofs are based on a result of Peligrad
and Utev (2006)concerning the asymptotic behavior of a class of linear processes
and spectral calculus.
KuelbsandZinndevelopcentrallimittheoremsforquantitleprocessesdefined
in terms of empirical processes of time dependent data. The key to their proofs is
an important extension of a method of Vervaat (1972).
