Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices

443 Pages·2011·2.23 MB·English

by  Göran Högnäs

Probability and Its Applications
PublishedinassociationwiththeAppliedProbabilityTrust
Editors:J. Gani,C.C. Heyde,P. Jagers,T.G.Kurtz
Forothertitlespublishedinthisseries,goto
www.springer.com/series/1560
About theAuthors:
Go¨ranHo¨gna¨sisProfessorofAppliedMathematicsathisalmamaterA˚boAkademi
University, from which he received his Ph.D. in Mathematics in 1974. Professor
Ho¨gna¨s is the co-author of a text on applied mathematics as well as co-editor
of the Proceedings of the Third Finnish-Soviet Syposium on Probability Theory
and Mathematical Statistics. He serves on the editorial board of the Journal of
TheoreticalProbability.ProfessorHo¨gna¨swasthedirectoroftheFinnishGraduate
SchoolinStochasticsandStatistics(1998–2009).
Arunava Mukherjea is Professor of Mathematics at the University of Texas–Pan
American, and prior to 2007, was Professor at the University of South Florida.
He received his Ph.D. in Mathematics in 1967 from Wayne State University. As
visitingprofessor,hespentsojournsatvariousacademicinstitutionsincludingTata
InstituteofFundamentalResearchinMumbai,IndiaandUniversiteofPaulSabatier
in Toulouse,France. He was a Fulbrightscholar. He is an associate editor (2005–
present)of the Journalof TheoreticalProbability,andwas its editor-in-chiefprior
to2005.
Go¨ran Ho¨gna¨s (cid:2) Arunava Mukherjea
Probability Measures
on Semigroups
Convolution Products, Random Walks,
and Random Matrices
ABC
Go¨ranHo¨gna¨s ArunavaMukherjea
A˚boAkademiUniversity UniversityofTexas–PanAmerican
Fa¨nriksgatan3B DepartmentofMathematics
20500A˚bo 1201WestUniversityDrive
Finland Edinburg,Texas78539
ghognas@abo.fi USA
[e-mail]
SeriesEditors:
SørenAsmussen PeterJagers
DepartmentofMathematicalSciences MathematicalStatistics
AarhusUniversity ChalmersUniversityofTechnology
NyMunkegade andGo¨teborg(Gothenburg)University
8000AarhusC 41296Go¨teborg
Denmark Sweden
[e-mail] [e-mail]
JoeGani ThomasG.Kurtz
CentreforMathematics DepartmentofMathematics
anditsApplications UniversityofWisconsin-Madison
MathematicalSciencesInstitute 480LincolnDrive
AustralianNationalUniversity Madison,WI53706-1388
Canberra,ACT0200 USA
Australia [e-mail]
[e-mail]
ISSN1431-7028
ISBN978-0-387-77547-0 e-ISBN978-0-387-77548-7
DOI10.1007/978-0-387-77548-7
LibraryofCongressControlNumber:2010938438
(cid:2)c SpringerScience+BusinessMedia,LLC2011
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein
connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,
orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Coverdesign:WMXDesign
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
From the Preface to the First Edition
“A Scientific American article on chaos, see Crutchfield et al. (1986),illustrates a
very persuasive example of recurrence. A painting of Henri Poincare´, or rather a
digitized versionof it, is stretched and cut to producea mildly distorted image of
Poincare´.Thesameprocedureisappliedtothedistortedimageandtheprocessisre-
peatedoverandoveragainonthesuccessivelymoreandmoreblurredimages.After
a dozen repetitionsnothing seems to be left of the originalportrait. Miraculously,
structuredimagesappearbrieflyaswecontinuetoapplythedistortionprocedureto
successiveimages.After241iterationstheoriginalpicturereappears,unchanged!
ApparentlythepixelsofthePoincare´ portraitweremovingaboutinaccordance
withastrictlydeterministicrule.Moreimportantly,thesetofallpixels,thewhole
portrait, was transformed by the distortion mechanism. In this example the trans-
formation seems to have been a reversible one since the original was faithfully
recreated.
It is not veryfarfetchedto introducea certain amountof randomnessand irre-
versibilityintheaboveexample.Thinkofarandommiscoloringofsomepixelsor
ofinadvertentlygivingapixelthecolorofitsneighbor.
Themethodsinthisbookaregearedtowardsbeingapplicabletotheasymptotics
ofsuchtransformationprocesses.Thetransformationsformasemigroupinanatu-
ralway;wewanttoinvestigatethelong-termbehaviorofrandomelementsofthis
semigroup.
To be more specific, let us consider a sequence of independentand identically
distributed random variables X ;X ;X ;::: taking values in a set of affine maps
0 1 2
fromRd into Rd, thatis, mapsof the formf.x/ D AxCB, whereB and x are
d(cid:3)1columnvectorsand(cid:2)Aisa(cid:3)d(cid:3)d realmatrix.Sincef canbeidentifiedwiththe
AB
.dC1/(cid:3).dC1/matrix ,therandomvariablesX scanalsoberegardedas
i
0 1
.d C1/(cid:3).d C1/randommatrices;thus,(cid:2),thedistributionofX(cid:2)i,isa(cid:3)probability
AB
measure on the set of .d C1/(cid:3).d C1/ matrices of the form . Let S be
0 1
theclosed(withusualtopology)multiplicativesemigroupgeneratedbythesupport
of(cid:2).ThenthestudyoftherandomwalksY ; Y DX X :::X withvaluesinS
n n 0 1 n
anddistribution(cid:2)n (thenthconvolutionpowerof(cid:2)),andthesetofrecurrentstates
v
vi FromthePrefacetotheFirstEdition
of .Y / become relevant in the context of the so-called iterated function systems
n
introducedbyBarnsleyandhiscolleagues[seeBarnsley(1988)].
Letusbrieflydiscussanotherexample.
Supposethatwearemonitoringarandomsystemwithtwostatesdenoted0and1.
Let
010001101001100010011001111000
and
101110010110011101111001111000
be observed time series of the successive states of the system. The observations
seemratherlikearecordofindependentcointosses,with0forheadsand1fortails,
say.ViewedasaMarkovchainonthetwo-statestatespaceX Df0;1gourprocess
wouldhavethetransitionprobabilitymatrix
(cid:2) (cid:3)
1=2 1=2
P D :
1=2 1=2
Letusassume,however,thattheabovetimeseriesareconcurrent.Thenanother
interpretationimposesitself:thestatespaceissubjectedtoasuccessionofrandom
transformations.(Thefirsttwotransformationaretranspositions,0and1justtrade
places.Atthethirdandfourthstepstheidentitymapisatwork.Asequenceoftrans-
positionsandidentitiesthenfollows,butatstep 19 everythingismappedontothe
state1.Fromthenonthetwopathsareidentical.)Thetransformationsarethefour
possible mappingsof X into itself, the identity (cid:3), the transposition(cid:4), and the two
constantmappings0and 1.Thetransitionmatrix P isthena convexcombination
ofmatricesrepresentingthosetransformations:
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
1 0 0 1 1 0 0 1
P Da Cb Cc Cd ;
0 1 1 0 1 0 0 1
wherea,b,c,darenonnegativenumberswithaCbCcCd D1.
Thus, a natural way to analyze our observed time series is to think of them as
emanating from an independent, identically distributed sequence of mappings of
thestatespaceintoitself,or,inotherwords,arandomwalkonthetransformations
ofX.
A Markov chain on a finite state space can always be regardedin this way. Its
transitionmatrixP isaconvexcombinationof0-1matricesrepresentingmappings
ofthestatespaceX intoitself.(IfP isdoublystochasticwecanwriteituniquely
as a convex combination of permutation matrices, this is the celebrated Birkhoff
theorem.)ThecorrespondingresultistrueevenforalargeclassofMarkovchains
onatopologicalspaceX,seeKifer(1986),Chapter1.
Toconsideranexampleinthecontextofparticlesystems,letV beanarbitrary
countably infinite set (with discrete topology),and let (cid:5) denote the semigroup of
FromthePrefacetotheFirstEdition vii
functionsf W V ! V undercomposition.Wecanthenidentifyeachf in(cid:5) with
aninfinite0-1stochasticmatrixA suchthat
f
.A / D1 ifandonlyif f.i/Dj:
f ij
By a configuration (cid:6) of V, we mean a nonnegativeinteger valued function on V
suchthat X
(cid:6).x/<1:
x2V
Theideaisthat(cid:6).j/isthenumberofparticlesthatoccupythesitej 2V,andthat
whenweapplythemappingf W V ! V,alltheseparticlesmovetothesitef.j/,
andtheconfigurationchangesto(cid:6)(cid:2)A ,where
f
X
.(cid:6)(cid:2)A /.x/D (cid:6).y/(cid:2)ı .x/:
f f.y/
y
Thenew configurationhasat site x all the particlesthatthe map f has sentto
the site x from the sites of the original configuration. Thus, to study the random
motionsoffinitesystemsofparticlesonV,withoutbirthsordeaths,whereeachsite
maybeoccupiedbyafinitenumberofparticles,andallparticlesataparticularsite
movetogether,oneneedstostudytherandomtransformationsF (thatis,theinfinite
randomstochasticmatricesA ).Insteadofstudyingthedifferentconfigurations,we
F
studyasequenceofindependentidenticallydistributedcountablyinfinitestochastic
matrices,andamongotherthings,willbeinterestedingainingsomeinsightsinthe
limitinglawsofproductsofthesematrices.
Tomentionyetanothercontextwhereprobabilitymeasuresoncountablesemi-
groupshavebeenfounduseful,wementionthepaperofHanselandPerrin(1983),
wheretheauthorsutilizedthestructureofanidempotentprobabilitymeasureona
semigroupinordertohavesomeinsightsincertainproblemsincodingtheory.
ItisalsorelevanttomentionthatRuzsa(1994)utilizedhisresultsonweak(cid:3)con-
vergenceofthesequence(cid:2) (cid:4)(cid:2) (cid:4):::(cid:4)(cid:2) ,wherethe(cid:2) sareprobabilitymeasures
1 2 n i
onacountablesemigroup,inprovingageneralizationofaresultinnumbertheory
duetoDavenportandErdo¨s(1936).Thislastmentionedresultsimplysaysthatev-
erymultiplicativeidealAofthesetN ofpositiveintegershasalogarithmicdensity,
thatis, X
1 1
(cid:7).A/D lim ;
n!1logn a
a 2A
a(cid:5)n
exists. Note that for a set A (cid:6) N, its logarithmic density (cid:7) may exist while its
asymptoticdensityd,givenby
X
1
d.A/D lim 1;
n!1n
a2A
a (cid:5)n
viii FromthePrefacetotheFirstEdition
may not exist. [It is well known, however, that (cid:7).A/ exists whenever d.A/ does,
andthen(cid:7).A/Dd.A/.]Ruzsa’sresultsaysthefollowing:iff isahomomorphism
fromthemultiplicativesemigroupofintegerstoacommutativesemigroupH,then
foreveryh2H,thesetfn2N W f.n/Dhghasalogarithmicdensity.
Letusfinallymention,beforewego tothe textproper,thatabstractsemigroup
theory was of crucial importance in developing the methods used in Ho¨gna¨s and
Mukherjea(1980)tostudythesetofrecurrentstatesofarandomwalktakingvalues
inn(cid:3)nrealmatrices.”
References
Barnsley,M.F.,FractalsEverywhere,AcademicPress,Orlando(1988).
Crutchfield,J.P.,J.D.Farmer,N.H.Packard,andR.S.Shaw,“Chaos,”ScientificAmerican255,
No.6,38–49(1986).
Gelbaum, B. R. and G. K. Kalisch, “Measures in Semigroups,” Canadian J. Math. 4, 396–406
(1952).
Hansel,G.andD.Perrin, “Codes andBernoullipartitions,”MathSystemsTheory 16,133–157
(1983).
Hennion,H.,“Limittheoremsforproductsofpositiverandommatrices,”TheAnnalsofProbability
25,No.4,1545–1587(1997).
Ho¨gna¨s,G.andA.Mukherjea,“Recurrentrandomwalksandinvariantmeasuresonsemigroupsof
n(cid:4)nmatrices,”MathZeitschrift173,69–94(1980).
Kifer, Y., Ergodic Theory of Random Transformations, Birkha¨user, Boston–Basel–Stuttgart
(1986).
Raugi, A., “Une demonstration du the´oreme de Choquet-Deny par les martingales,” Ann. Inst.
HenriPoincare´Statist.Probab.19,101–109(1983).
Ruzsa,I.Z.,Logarithmicdensityandmeasuresonsemigroups,ManuscriptaMathematica89,307–
317(1996).
Preface to the Second Edition
Inthisnewedition,wehavecorrectedallerrors(mathematicalandotherwise)that
wecouldfindorwerebroughttoourattentionfromthebook’sfirsteditionbyour
colleagues and students. We have also updated the references, all notes and com-
mentsattheendofeachchapter,andmoreimportantly,addedexercisesattheend
ofeach sectionin each chapter.We have also addednew resultsoftenin the main
textandintheappendices.
Thebook,wefeel,canbeausefulreferenceforcoursesintheareaofprobability
onalgebraicstructures.Itcanalsoserveasatextforgraduateorseniorundergrad-
uatestudentsforanonesemestercourseon“Probabilitymeasuresonsemigroups”
designedaroundthefollowingfourcoretopics:
(i) Completely simple semigroups: their structure and the Rees-Suschkewitsch
theorem(seeChapter1)
(ii) Convolutionproductsofprobabilitymeasuresonlocallycompactsemigroups,
theirweakconvergence,andtheirweaklimitswhichareidempotentprobabil-
itymeasures,withsupportsalwaysclosedcompletelysimplesubsemigroups,
andstructuretheoremforidempotentprobabilitymeasures(seeChapter2)
(iii) Recurrentrandomwalksonsemigroups:characterizationofthesetofrecurrent
statesasclosedandcompletelysimplesubsemigroups(seeChapter3)
(iv) Tightnessandconvergenceindistributionofproductsofdbydi.i.d.random
nonnegativematrices: The Kesten-Spitzer theoryand variousgeneralizations
torandomrealmatrices(seeChapter4)
Thefirstchapterofthebookcanalsoserveasatextforanonesemestercourse
on Semigroups for senior undergraduate or graduate students. Sufficiently many
exercisesofvaryinglevelsofdifficultyhavebeenincludedtohelptheinstructorof
suchacourse.
Semigroupsareverygeneralstructuresandweoftencomeacrosstheminvarious
contextsinscienceandengineering.(Seetheprefacetothefirstedition.)Theresults
thatwehavepresentedhereonweakconvergenceorrandomwalksorrandomma-
tricesusingsemigroupideasareforthemostpartcompleteandbestpossible.Still
ix