Table Of ContentSpringer Monographs in Mathematics
FromRobert
For my parents, Kurt and Berta Adler, who taught me the most important lesson
ofall:
Knowfromwhenceyouhavecome,towhereyouaregoing,
andbeforeWhomyoumustultimatelyreport.
EthicsoftheFathers,Chapter3
FromJonathan
For my wife, Lee-Ann, and daughter, Isobel, for their tremendous support of my
effortsinwritingthisbookandmyparents,JohnandBrenda,foralloftheirencour-
agement.
Robert J. Adler
Jonathan E. Taylor
Random Fields and Geometry
With 21 Illustrations
Robert Adler Jonathan Taylor
Faculty of Industrial Engineering Department of Statistics
and Management Stanford University
Technion – Israel Institute of Technology Sequoia Hall
Haifa,32000 Stanford,CA 94305-4065
Israel U.S.A.
Mathematics Subject Classification (2000):60-02,60Gxx,60G60,53Cxx,53C65
Library ofCongress Control Number:2007923465
ISBN-13:978-0-387-48112-8 e-ISBN-13:978-0-387-48116-6
Printed on acid-free paper.
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Preface
Sincetheterm“randomfield’’hasavarietyofdifferentconnotations,rangingfrom
agriculture to statistical mechanics, let us start by clarifying that, in this book, a
randomfieldisastochasticprocess,usuallytakingvaluesinaEuclideanspace,and
definedoveraparameterspaceofdimensionalityatleast1.
Consequently,randomprocessesdefinedoncountableparameterspaceswillnot
appear here. Indeed, even processes on R1 will make only rare appearances and,
fromthepointofviewofthisbook,arealmosttrivial. Theparameterspaceswelike
bestaremanifolds,althoughformuchofthetimeweshallrequirenomorethanthat
theybepseudometricspaces.
With this clarification in hand, the next thing that you should know is that this
bookwillhaveasequeldealingprimarilywithapplications.
In fact, as we complete this book, we have already started, together with KW
(KeithWorsley),onacompanionvolume[8]tentativelyentitledRFG-A,orRandom
Fields and Geometry: Applications. The current volume—RFG—concentrates on
thetheoryandmathematicalbackgroundofrandomfields,whileRFG-Aisintended
to do precisely what its title promises. Once the companion volume is published,
youwillfindtherenotonlyapplicationsofthetheoryofthisbook,butof(smooth)
randomfieldsingeneral.
Makingaclearsplitbetweentheoryandpracticehasbothadvantagesanddisad-
vantages. Itcertainlyeasedthepressureonustoattemptthealmostimpossiblegoal
ofwritinginastylethatwouldbeaccessibletoall. Italso,toalargeextent,easesthe
loadonyou,thereader,sinceyoucannowchoosethevolumeclosertoyourinterests
andsoavoideither“irrelevant’’mathematicaldetailorthe“realworld,’’dependingon
youroutlookandtastes. However,thesearesmallgainswhencomparedtothemajor
lossofcreatinganapparentdichotomybetweentwothingsthatshould,inprinciple,
gohand-in-hand: theoryandapplication. Whatistrueinprincipleisparticularlytrue
ofthetopicathand,and,toexplainwhy,weshallindulgeourselvesinaparagraph
ortwoofhistory.
Theprecusortobothofthecurrentvolumeswasthe1981monographTheGeom-
etryofRandomFields(GRF)whichgrewoutofRJA’s(i.e.,RobertAdler’s)Ph.D.
thesisunderMichaelHasofer. Theproblemthatgavebirthtothethesiswasanapplied
vi Preface
one,havingtodowithgroundfissuresduetowaterpermeatingthroughtheearthun-
derabuildingsite. However,boththethesisandGRF endedupbeingdirectedmore
to theoreticians than to subject-matter researchers. Nevertheless, the topics there
foundmanyapplicationsoverthepasttwodecades,indisciplinesaswidespreadas
astrophysicsandmedicalimaging.
These applications led to a wide variety of extensions of the material of GRF,
which,whiledifferentinextenttowhatwasthere,werenotreallydifferentinkind.
However, in the late 1990s KWfound himself facing a brain mapping problem on
the cerebral cortex (i.e., the brain surface) that involved looking at random fields
onmanifolds. JonathanTaylor(JET)lookedatthisproblemand,insomewhatofa
repetitionofhistory,tookittoanabstractlevelandwroteaPh.D.thesisthatcompletely
revolutionized1 the way one should think about problems involving the geometry
generatedbysmoothrandomfields. This,andsubsequentmaterial,makesupPartIII
ofthecurrent,three-part,book.
In fact, this book is really about Part III, and it is there that most of the new
material will be found. Part I is mainly an adaptation of RJA’s 1990 IMS lecture
notes,AnIntroductiontoContinuity,Extrema,andRelatedTopicsforGeneralGauss-
ianProcesses,considerablycorrectedandsomewhatreworkedwiththeintentionof
providingallthatoneneedstoknowaboutGaussianrandomfieldsinordertoread
Part III. In addition, Part I includes a chapter on stationarity. En passant, we also
included many things that were not really needed for Part III, so that Part I can be
(andoftenhasbeen)usedasthebasisofaone-quartercourseinGaussianprocesses.
Suchacourse(and, indeed, thisbookasawhole)wouldbeaimedatstudentswho
havealreadytakenabasiccourseinmeasure-theoreticprobabilityandalsohavesome
basicfamiliaritywithstochasticprocesses.
PartIIcoversmaterialfrombothintegralanddifferentialgeometry. However,the
materialhereisconsiderablylessstandardthanthatofPartI,andweexpectthatfew
readersotherthanprofessionalgeometerswillbefamiliarwithallofit. Inaddition,
someoftheproofsaredifferentfromwhatisfoundinthestandardgeometryliterature
inthattheyusepropertiesofGaussiandistributions.2
TherearetwomainaimstoPartII. Oneistosetupananalogueofthecriticalpoint
theory of Marston Morse in the framework of Whitney stratified manifolds. What
makesthisnonstandard(atleastintermsofwhatmoststudentsofmathematicssee
aspartoftheirgraduateeducation)isthatMorsetheoryisusuallydoneforsmooth
manifolds, preferably without boundaries. Whitney stratified manifolds are only
piecewisesmooth,andarepermittedanynumberofedges,corners,etc. Thisbrings
themclosertotheobjectsofintegralgeometry,towhichwedevoteachapter. While
the results of this specific chapter are actually subsumed by what we shall have to
sayaboutWhitneystratifiedmanifolds,theyhavetheadvantagethattheyareeasyto
stateandprovewithoutheavymachinery.
ThesecondaimofPartIIistodevelopLipschitz–Killingcurvaturesinthesetting
ofWhitneystratifiedmanifoldsandtodescribetheirroleinwhatareknownas“tube
1ThisverbwaschosenbyRJAandnotJET.
2Afterall,sinceweshallbythenhavetheGaussianPartIbehindus,itseemswastefulnot
touseitwhenitcanhelpsimplifyproofs.
Preface vii
formulas.’’ We shall spend quite some time on this. Some of the material here is
“wellknown’’(albeitonlytoexperts)andsome,particularlythatrelatingtotubeand
CroftonformulasinGaussspace,isnew. Furthermore,wederivethetubeformulas
forlocallyconvexWhitneystratifiedmanifolds,whichisbothsomewhatmoregeneral
thantheusualapproachforsmoothmanifolds,andsomewhatmorepractical,since
mostoftheparameterspacesweareinterestedinhaveboundaries. Inaddition,the
approachweadoptisoftenunconventional.
Thesetwoaimsmakeforasomewhatunusualcombinationofmaterialandthereis
noeasilyaccessibleandsuccinct3alternativetoourPartIIforlearningaboutthem. In
thesamevein,inordertohelpnovicedifferentialgeometers,wehaveincludedaone-
chapterprimerondifferentialgeometrythatrunsquickly,andoftenunaesthetically,
throughthebasicconceptsandnotationofthismostbeautifulpartofmathematics.
However,althoughPartsIandIIofthisbookcontainmuchmaterialofintrinsic
interestwewouldnothavewrittenthemwereitnotforPartIII,forwhichtheyprovide
necessarybackgroundmaterial. WhatisitinPartIIIthatjustifiescloseto300pages
of preparation? Part III revolves around the excursion sets of smooth, Rk-valued
randomfieldsf overpiecewisesmoothmanifoldsM. Excursionsetsaresubsetsof
M givenby
A ≡A (f,M)=(cid:2) {t ∈M :f(t)∈D}
D D
forD ⊂Rk.
Agreatdealofthesamplefunctionbehaviorofsuchfieldscanbededucedfrom
theirexcursionsetsandasurprisingamountfromtheEuler,orEuler–Poincaré,char-
acteristicsoftheseexcursionsets, definedinPartII. Inparticular, ifwedenotethe
EulercharacteristicofasetAbyϕ(A), thenmuchofPartIIIisdevotedtofinding
thefollowingexpressionfortheirexpectation,whenf isGaussianwithzeromean
andunitconstantvariance:
d(cid:2)imM
E{ϕ(A )}= (2π)−j/2L (M)Mk(D). (0.0.1)
D j j
j=0
HeretheL (M)aretheLipschitz–KillingcurvaturesofMwithrespecttoaRiemann-
j
ianmetricinducedbytherandomfieldf,andtheMk(D)arecertainMinkowski-like
j
functionals (closely akin to Lipschitz–Killing curvatures) on Rk under Gauss mea-
sure.
Ifallofthissoundsterriblyabstract, thetruthisthatitbothis, andisnot. Itis
abstract,becausewhile(0.0.1)hashadmanyprecursorsoverthelast60yearsorso,it
hasneverbeforebeenestablishedinthegeneralitydescribedabove. Itisalsoabstract
inthatthetoolsinvolvedinthederivationof (0.0.1)inthissettingrequiresomerather
heavymachineryfromdifferentialgeometry. However,thislevelofabstractionhas
3Thestresshereison“succinct.’’WiththeexceptionofthematerialonGaussspace,almost
everythingthatwehavetosaycanbefoundsomewhereintheliteraturesofintegraland
differentialgeometry,forwhichtherearemanyexcellenttexts,someofwhichweshalllist
later. However,allpresumeabackgroundknowledgethatisbeyondwhatweshallrequire,
andeachcontainsonlyasubsetoftheresultsweshallneed.
viii Preface
turnedouttopaysignificantdividends,fornotonlydoesityieldinsightintoearlier
resultsthatwedidnothavebefore,butitalsohaspracticalimplications. Forexample,
theapproachthatweshallemployworksjustaswellfornonstationaryprocessesas
itdoesforstationaryones.4 However,nonstationarity,evenonmanifoldsassimple
as[0,1]2,waspreviouslyconsideredessentiallyintractable. Simplyput,thisisone
of those rare but constantly pursued examples in mathematics in which abstraction
leadsnotonlytoacompleteandeleganttheory,butalsotopracticalconsequences.
Anextremelysimpleandverydown-to-earthapplicationof (0.0.1)ariseswhen
the manifold is the unit interval [0,1], f is real-valued, and D = [u,∞). In that
case,E{ϕ(A )}isnomorethanthemeannumberofupcrossingsofthelevelubythe
D
processf,alongwithaboundarycorrectionterm. Consequently,modulothebound-
ary term, (0.0.1) collapses to no more than the famous Rice formula, undoubtedly
oneofthemostimportantresultsintheapplicationsofsmoothstochasticprocesses.
IfyouareunfamiliarwithRice’sformula,thenyoumightwanttostartreadingthis
book at Section 11.1, where it appears in some detail, together with heuristic, but
instructional,proofsandapplications.
OneofthereasonsthatRice’sformulaissoimportantisthatithaslongbeenused
asanapproximation,forlargeu,totheexcursionprobability
(cid:3) (cid:4)
P sup f(t)≥u ,
t∈[0,1]
itself an object of major practical importance. The heuristic argument behind this
is simple: If f crosses a high level, it is unlikely do so more than once. Thus, in
essence,theprobabilitythatf crossestheleveluisclosetotheprobabilitythatthere
isanupcrossingofu,alongwithaboundarycorrectionterm. (Thecorrectioncomes
from the fact that one way for supt∈[0,1]f(t) to be larger than u is for there to be
noupcrossingsbutf(0)≥u.) Sincethenumberofupcrossingsofahighlevelwill
alwaysbesmall,theprobabilityofanupcrossingiswellapproximatedbythemean
numberofupcrossings. HenceRice’sformulagivesanapproximationforexcursion
probabilities.
If (0.0.1)isthemainresultofPartIII,thenthesecond-most-importantresultis
that,atthesamelevelofgeneralityandforawidechoiceofD,wecanfindabound
forthedifference
|P{∃t ∈M :f(t)∈D}−E{ϕ(A )}|.
D
Aspecificcaseofthisoccurswhenf takesvaluesinR1,inwhichcasenotonlycan
weshowthat,forlargeu,
(cid:5) (cid:3) (cid:4) (cid:5)
(cid:5) (cid:5)
(cid:5)P supf(t)≥u −E{ϕ(A[u,∞))}(cid:5)
t∈M
issmall,butwecanprovideanupperboundtoitthatisbothsharpandexplicit.
Giventhatthesecondtermhereisknownfrom(0.0.1),whatthisinequilitygivesis
anexcellentapproximationtoGaussianexcursionprobabilitiesinaverywidesetting,
somethingthathaslongbeenaholygrailofGaussianprocesstheory.
4Stillassumingmarginalstationarity,i.e.,zeromeanandconstantvariance.
Preface ix
Inthegeneralityinwhichweshallbeworking,theboundisdeterminedbygeo-
metricpropertiesofthemanifoldM withtheinducedRiemannianmetricmentioned
above. Furthermore,unlikethehandwavingargumentdescribedaboveforthesimple
one-dimensionalcase,thenewtoolsprovideafullyrigorousproof.
At this point we should probably say something about why we chose to take
piecewise smooth manifolds as our generic parameter space. This is perhaps best
explainedviaanexample. Supposethatweweretotakeparameterspacesthatwere
∞ ∞
smooth,evenC ,manifolds,withC boundaries.
C
A B
C
A
B
∞ ∞
Fig.0.0.1.AC functiondefinedoveramanifoldwithaC boundarygivesexcursionsets
thathavesharp,nondifferentiable,corners.
SuchanexampleisshowninFigure0.0.1,wheretheparameterspaceisadisk.
∞
Thethreeexcursionsofa(C )functionf abovesomenominallevelaremarkedon
thefunctionsurface,andtheselieabovethethreecorrespondingcomponentsofthe
excursionsetA[u,∞). Notethat, despitethesmoothnessofeachcomponentofthis
example,theexcursionsethassharpcornerswhereitintersectswiththeboundaryof
theparameterspace. Inotherwords,A[u,∞)isonlyapiecewisesmoothmanifold.
Itturnsoutthatsinceweendupwithpiecewisesmoothmanifoldsforourexcursion
sets,thereisnotalotsavedbynotstartingwiththemasparameterspacesaswell.5
So now you know what awaits you at the end of the path through this book.
However,traversingthepathhasvalueinitself. Wandering,asitdoes,throughthe
fieldsofbothprobabilityandgeometry,itisapaththatweimaginenottoomanyof
you will have traversed before. We hope that you will enjoy the scenery along the
wayasmuchaswehaveenjoyeddescribingit. (Wealsohope,foryoursake,thatit
willbeeasierandfasterinthereadingthanitwasinthewriting.)
Weareleftnowwithtwotasks:Advisinghowbesttoreadthisbook,andoffering
ouracknowledgments.
5Ofcourse,wecouldsimplifythingsconsiderablybyworkingonlywithparameterspaces
that have no boundary, something that would be natural, for example, for a differential
geometer. However, this would leave us with a theory that could not handle parameter
spacesassimpleasthesquareandthecube,asituationthatwouldbeintolerablefromthe
pointofviewofapplications.
x Preface
The best way to read the book is, of course, to start at the beginning and work
throughtotheend. Thatwashowwewroteit. However,heresomeotherpossibilities,
dependingonwhatyouwanttogetoutofit.
(i) AcourseonGaussianprocesses: Chapters1through4alongwithSections5.1
through5.4ifyouwanttolearnaboutstationarityaswell. Thesechapterscan
bereadinmoreorlessanyorder;seethecommentsintheintroductiontoPartI.
(ii) RandomfieldsonEuclideanspaces,withanaccentongeometry: Sections1.1–
1.4.2 and Chapter 3 for basic Gaussian processes, Sections 4.1, 4.5, and 4.6
forsomeclassicalmaterialonextremaldistributions,andChapter5onstation-
arity. Chapter 6 and Section 9.4 give the basic geometry and Chapter 11 the
randomgeometryofGaussianfields. Section14.4givesexamplesofhowthe
resultsofChapter11relatetoexcursionprobabilities,andSection15.10gives
examples of the non-Gaussian theory. (Note that because you have chosen to
remain in the Euclidean scenario, and so avoid most of the real challenges of
differential geometry, you have been relegated to reading examples instead of
thegeneralcase!)
(iii) Probabilisitic problems in, and using, differential geometry: Sections 1.1, 1.2
andtheresults(butnotproofs)ofChapter3togetabare-bonesintroductionto
Gaussianprocesses,alongwithSections5.5and5.6forsomeimportantnotation.
As much of Chapter 7 as you need to revise differential-geometric concepts,
followedbyChapters8,9,and10. ThepunchlineistheninChapters12and13
forGaussianprocessesandChapter15ingeneral. Itisonlyinthislastchapter
thatyouwillgettoseeallthegeometricpreliminariesofPartIIinplayatonce.
(iv) Applicationswithoutthetheory: WaitforRFG-A.Weareworkingonit!
Nowfortheacknowledgments. BothRJAandJETowedebtsofgratitudetoKW,
andwehadbetteracknowledgethemnow, sincewecanhardlydoitinthepreface
ofRFG-A.
Beyond our personal debts to KW, not least for getting the two us of together,
the subject matter of this book also owes him an enormous debt of gratitude. It
was during his various extensions and applications of the material of GRF that the
passagebetweentheoldEuclideantheoryanditsnewermanifoldversionbeganto
take shape. Without his tenacious refusal to leave (applied) problems because the
theory (geometry) seemed too hard, the foundations on which our Part III is based
wouldneverhavebeenlaid.
Backtothepersonallevel,wealsoowedebtsofgratitudetonumerousstudents
attheTechnion,UCSantaBarbara,Stanford,andtheICE-EMinBrisbanewhosat
throughcoursesasweputthisvolumetogether,aswellasthegroupatMcGillthat
wentthroughthebookasareadingcoursewithKW.Theirenthusiasm,patience,and
refusaltotake“itiseasytoseethat’’forananswerwhenitwasnotallthateasyto
seethings,nottomentionallthetyposanderrorsthattheyfound,hashelpedirona
lotofwrinklesoutofthefinalproduct.
In particular, we would like to thank Nicholas Chamandy, Sourav Chatterjee,
SteveHuntsman,FarzanRohani,AlessioSancetta,ArminSchwartzman,andSreekar
Vadlamani for their questions, comments, and, embarrassingly often, corrections.
