The Generic Chaining: Upper and Lower Bounds for Stochastic Processes

225 Pages·2005·1.348 MB·English

by  Michel Talagrand

S M M
pringer onographs in athematics
Michel Talagrand
The Generic
Chaining
Upper and Lower Bounds
of Stochastic Processes
123
MichelTalagrand
Equiped’Analysefonctionelle
InstitutMathématiquesdeJussieu
U.M.R.7586C.N.R.S.
UniversitéParisVI
Boite186
4PlaceJussieu75230Paris,France
and
TheOhio-StateUniversity
281West18thAvenue
Columbus,OH43210,USA
e-mail:[email protected]
LibraryofCongressControlNumber:2005920702
MathematicsSubjectClassification(2000):60G15,60G17,60G52,60D05,60G70,60B11
ISSN1439-7382
ISBN3-540-24518-9SpringerBerlinHeidelbergNewYork
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To my father
Contents
Introduction.................................................. 1
1 Overview and Basic Facts................................. 5
1.1 Overview of the Book ................................... 5
1.2 The Generic Chaining................................... 9
1.3 A Partitioning Scheme .................................. 17
1.4 Notes and Comments ................................... 30
2 Gaussian Processes and Related Structures ............... 33
2.1 Gaussian Processes and the Mysteries of Hilbert Space ...... 33
2.2 A First Look at Ellipsoids ............................... 39
2.3 p-stable Processes ...................................... 42
2.4 Further Reading: Stationarity ............................ 50
2.5 Order 2 Gaussian Chaos................................. 51
2.6 L2, L1, L∞ Balls ...................................... 60
2.7 Donsker Classes ........................................ 75
3 Matching Theorems ...................................... 89
3.1 The Ellipsoid Theorem.................................. 89
3.2 Matchings ............................................. 94
3.3 The Ajtai, Komlo`s, Tusn`ady Matching Theorem............ 97
3.4 The Leighton-Shor Grid Matching Theorem. ............... 102
3.5 Shor’s Matching Theorem ............................... 110
4 The Bernoulli Conjecture................................. 129
4.1 The Conjecture ........................................ 129
4.2 Control in (cid:1)∞ Norm .................................... 130
4.3 Chopping Maps and the Weak Solution ................... 134
4.4 Further Thoughts ...................................... 144
5 Families of distances...................................... 145
5.1 A General Partition Scheme ............................. 145
5.2 The Structure of Certain Canonical Processes.............. 150
5.3 Lower Bounds for Infinitely Divisible Processes............. 160
VIII Contents
5.4 The Decomposition Theorem
for Infinitely Divisible Processes.......................... 175
5.5 Further Thoughts ...................................... 181
6 Applications to Banach Space Theory .................... 185
6.1 Cotype of Operators from C(K).......................... 185
6.2 Computing the Rademacher Cotype-2 Constant ............ 196
6.3 Restriction of Operators................................. 202
6.4 The Λ(p) Problem...................................... 207
6.5 Schechtman’s Embedding Theorem ....................... 213
6.6 Further Reading........................................ 216
References.................................................... 217
Index......................................................... 221
Introduction
What is the maximum level a certain river is likely to reach over the next
25 years? (Having experienced three times a few feet of water in my house,
I feel a keen personal interest in this question.) There are many questions of
the samenature:whatisthe likelymagnitude ofthe strongestearthquaketo
occurduringthelifeofaplannedbuilding,orthespeedofthestrongestwind
a suspension bridge will have to stand? All these situations can be modeled
in the same manner. The value X of the quantity of interest (be it water
t
level or speed of wind) at time t is a random variable. What can be said
about the maximum value of X over a certain range of t?
t
A collectionof random variables (X ), where t belongs to a certain index
t
set T, is called a stochastic process, and the topic of this book is the study
of the supremum of certain stochastic processes, and more precisely to find
upper and lower bounds for the quantity
EsupX . (0.1)
t
t∈T
Since T might be uncountable, some care has to be taken to define this
quantity. For any reasonable definition of Esup X we have
t∈T t
EsupX =sup{EsupX ; F ⊂T , F finite}, (0.2)
t t
t∈T t∈F
an equality that we will take as the definition of the quantity Esup X .
t∈T t
Thus, the crucial case for the estimation of the quantity (0.1) is the case
where T is finite, an observation that should stress that this book is mostly
about inequalities.
The most important random variables (r.v.) are arguably Gaussian r.v.
The study of conditions under which Gaussian processes are bounded (i.e.
the quantity(0.1)isfinite)goesbackatleasttoKolmogorov.Thecelebrated
Kolmogorov conditions for the boundedness of a stochastic process are still
useful today, but they are far from being necessary and sufficient. The un-
derstanding of Gaussian processes was long delayed by the fact that in the
most immediate examples the index set is a subset of R or Rn and that the
temptation to use the specialstructure of this index set is nearly irresistible.
ProbablythesinglemostimportantconceptualprogressaboutGaussianpro-
cesses is the realization, in the late sixties, that the boundedness of a (cen-
tered) Gaussian process is determined by the structure of the metric space
2 Introduction
(T,d), where the distance d is given by
d(s,t)=(E(X −X )2)1/2 . (0.3)
s t
In1967,R.Dudley obtainedasharpsufficientconditionfortheboundedness
of a Gaussianprocess,the so-calledDudley entropy condition. It is basedon
the fact that, for a Gaussian process,
(cid:1) (cid:2)
u2
∀u>0, P(|X −X |≥u)≤2exp − . (0.4)
s t 2d(s,t)2
Dudley’s condition is however not necessary. A few years later, X. Fernique
(building on earlier ideas of C. Preston) introduced a condition based on
the use of a new tool called majorizing measures. Fernique’s condition is
weaker than Dudley’s, and Fernique conjectured that his condition was in
fact necessary. Gilles Pisier suggested in 1983 that I should work on this
conjecture, and kept goading me until I proved it in 1985, obtaining thus a
necessaryandsufficientconditionfortheboundednessofaGaussianprocess,
or equivalently, upper and lower bounds of the same order for the quantity
(0.1) in terms of the structure of the metric space (T,d). A few years of
great excitement followed this discovery, during which I proved a number
of extensions of this result, or of parts of it, to other classes of processes.
I was excited because I liked (and still like) these results. Unfortunately, I
was about the only one to get excited. Part of the reason is that Fernique’s
concept of majorizing measures is very difficult to grasp at the beginning,
and was consequently dismissed by the main body of probabilists as a mere
curiosity. (I must admit that I did have a terrible time myself to understand
it.)
In 2000, while discussing one of the open problems of this book with K.
Ball (be he blessed for his interest in it) I discovered that one could replace
majorizingmeasuresbyasuitablevariationontheusualchainingarguments,
a variation that is moreover totally natural. That this was not discovered
much earlier is a striking illustration of the inefficiency of the human brain
(and of mine in particular). This new approach not only removes the psy-
chological obstacle of having to understand the somewhat disturbing idea of
majorizing measures, it also removes a number of technicalities, and allows
onetogivesignificantlyshorterproofs.Ithusfeltthetimehadcometomake
anewexpositionofmybody ofworkonlowerandupperboundsforstochas-
tic processes. The feeling that, this time, the approach was possibly (and
even probably) the correct one gave me the energy to rework all the proofs.
For several of the most striking results, such as Shor’s matching theorem,
the decomposition theorem for infinitely divisible processes, and Bourgain’s
solution of the Λ problem, the proofs given here are at least three times
p
shorter than the previously published proofs.
Beside enjoying myself immensely and giving others a chance to under-
standtheresultspresentedhere(andevenpossiblytogetexcitedaboutthem)
Introduction 3
a main objective of this book is to point out several problems that remain
open. Of course opinions differ as to what constitutes an important prob-
lem, but I like those presentedhere. One of them deals with the geometryof
Hilbert space, a topic that can hardly be dismissed as exotic. I stated only
the problems that I find really interesting. Possibly they are challenging. At
least, I made every effort to make progress on them. A significant part of
the material of the book was discovered while trying to solve the “Bernoulli
problem” of Chapter 4. I have spent many years thinking to that problem,
and will be glad to offer a prize of $ 5000 for a positive solution of it. A
smaller prize of $ 1000 is offered for a positive solution of the possibly even
more important problemraisedat the end of chapter 5.The smaller amount
simply reflects the fact that I have spent less time on this question than on
the Bernoulli problem. It is of course advisable to claim these prizes before
I am too senile to understand the solution, for there will be no guarantee of
payment afterwards. (Cash awards will also be given for a negative solution
of any of these two problems, the amount depending on the beauty of the
solution.)
It is my pleasure to thank the Ohio State University and the National
Science Foundation for supporting the typing of this book, and making its
publication possible.
Imustapologizeforthecountlessinaccuraciesandmistakes,smallorbig,
that this book is bound to contain despite all the efforts made to remove
them. I was very much helped in this endeavor by a number of colleagues,
and in particular by A. Hanen and R. Latala, who read the entire book. Of
course, all the remaining mistakes are my sole responsibility.
Inconclusion,abitofwisdom.IthinkthatIfinallydiscoveredafoolproof
way to ensure that the writing of a book of this size be a delightful and easy
experience. Just write a 600 page book first!