Table Of ContentPaul Lévy and Maurice Fréchet
Sources and Studies in the History of
Mathematics and Physical Sciences
Forfurthervolumes:
www.springer.com/series/4142
Marc Barbut (cid:2) Bernard Locker (cid:2) Laurent Mazliak
Paul Lévy and
Maurice Fréchet
50 Years of Correspondence in 107 Letters
Translation of the introductory sections and most of the
notes by Peter Kleban, Professor Emeritus LASST &
Department of Physics and Astronomy University of Maine
MarcBarbut(deceased) LaurentMazliak
ProfessorBarbutwrotethiswhileatEHESS LaboratoiredeProbabilitésetModèles
Paris,France Aléatoires
UniversitéPierreetMarieCurie
BernardLocker Paris,France
UFRdeMathématiques
UniversitéRenéDescartes
Paris,France
ISSN2196-8810 ISSN2196-8829(electronic)
SourcesandStudiesintheHistoryofMathematicsandPhysicalSciences
ISBN978-1-4471-5618-5 ISBN978-1-4471-5619-2(eBook)
DOI10.1007/978-1-4471-5619-2
SpringerLondonHeidelbergNewYorkDordrecht
LibraryofCongressControlNumber:2013955594
MathematicsSubjectClassification: 01A60,60-03,60J65,31-03
BasedonatranslationfromtheFrenchlanguageedition:
‘PaulLévy-MauriceFréchet:50ansdecorrespondancemathématique’
editedbyMarcBarbut,BernardLockerandLaurentMazliak
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Tenyearsafterthepublicationofthefirstedition
inFrench,andthoughoneofusisnowamongus
onlyinspirit,itisstillthreegenerationswhoare
gladtodedicatethisbooktoourfriend,
colleagueandmaster
BernardBru
forhisseventiethbirthday,whoseenergyand
immenseculturehadalwaysbeenforusa
pricelesssupportandwhoseworkhas
contributedsoessentiallytothedisseminationof
thehistoryofmathematics.
MarcBarbut†,BernardLockerand
LaurentMazliak
Inmemoriam
SincetheFrenchversionofthisbookwas
published,ourcolleagueandfriendMarcBarbut
passedawayinDecember2011.Hisenthusiasm,
hisendlessculture,hispassionfortheworkof
thetwoprotagonists(hehadmetbothofthemin
hisyouth),wereessentialingredientsforthe
completionofthiswork.Wearegladtoseizethis
opportunitytorecallhisgoodmemoryandalso
toofferabookheappreciatedalottoalarger
audience.
BernardLockerandLaurentMazliak
Paris,September2013
Preface by Kai Lai Chung
PaulLévy’sbookswerenoteasyofaccessinKunming,China,duringthewaryears
attheSouthwestAssociatedUniversities,owingtoJapanesebombingofthecity.1
IarrivedatPrincetoninDecember1945butitwaslaterthatIsawCaptainGilHunt
poringoverl’Addition.2HetoldmethatvonNeumannsaid:readingLévy,onesoon
realizesthattheauthorthinksmathematicsinadifferentwayfromnousautres.
IfirstsawLévyatBerkeley,summerof1950,whenhewasgivingacoursewhich
I attended. One day I went to his office and asked him about a result for Markov
chains. At once he indicated a counter-example. It must be then or soon after that
I mentioned the conjecture regarding the positivity of a transition function p (t):
ij
eitheritvanishesforallt oritneverdoes.Inhisepoch-makingpaper3hebeganwith
thisoccasion.Hecallsastatei“stable”if−p(cid:2) (0)<∞,“instantaneous”otherwise.
ii
Abasicpropertyofastablestatehestates,isthatthetime-setinwhichastablestate
occurs is (almost surely) a sequence of disjoint intervals of which the cardinality
is finite up to any finite time. This must be obvious to him but missed by Mon-
sieur Doob who published two important papers on denumerable Markov chains.
The proof actually needs a little “measure-theoretic”fiddling which I presented at
Feller’sseminarlater.TheproofinLévy’spaperofthetruthoftheconjectureabove
isvalidifallstatesarestable,asconfirmedinthedoctoratethesisofR.V.Chacon
(atSyracuseUniversity).ItwastherecentlydeceasedDonaldG.Austinwhogave
a beautiful complete proof of the conjecture by so-called “path-analysis.”4 It was
likeLévythatheannouncedhisoversightinaComptesRendusnotesoonafterhe
recognizedit.HewrotemetosaythathelikedAustin’sproof.
Turningtheclockbackto1947-9,whenIwasatCornell,thediscoveryofLévy’s
long paper,5 not available during the war, caused a sensation with Feller and Kac.
1SeeSem.Stoch.Process.(1988).
2ThebookLévy(1937).
3Ann.Sci.Ec.Norm.Super.68:327–381(1951).
4SeeChung:Markovchains,Springer(1960&1967).
5Compos.Math.7:283–339(1939).
vii
viii PrefacebyKaiLaiChung
Among the numerous new results, the “arcsin” law together with its complement
“uniformlaw”forBrownianMovementwasanunheraldedevent.FellerandIwrote
a note transferring both results to the classical setting of coin-tossing, heads or
tails with fifty-fifty chance. The resulting statistical interpretation is so contrary
to common-sense “intuition” that had Chevalier de Méré been at the gaming ta-
ble he would certainly have rushed to consult Fermat or Pascal. Indeed the news
wasevenreportedinapopularsciencebulletin,ofwhichFellerkiddedmeasbeing
theinformer—Iwasnot.Lévy’sarticlecontainssuchawealthofideasandmethods
whichremaintobeexploredtothisday.
Probably the best known of Lévy’s multifarious achievements is the family of
“infinitely divisible laws” with numerous ramifications and “applications” includ-
ing the “normal” and “stable” laws (another kind of stability, both named by the
discoverer). The general formula for their Fourier transform (ch.f.) is sometimes
citedas“Khinchine-Lévy,”insillyEnglishalphabeticalorder.6InaletterLévysaid
K.completeditintheunsymmetricCauchycase(only).InthebookbyGnedenko
andKolmogorov,itwasproperlynamed.WhatisnotwellknownisthatinLévy’s
hands, it is just the analytic portion of his profound study of the paths of “Lévy
process(es),” formerly called “additive,” “differential” etc. I must refer the really
interested reader to my Review of Doob’s book Stochastic Processes7 for further
comments.LetmeaddthatifLévy’sotherrenownedpublicationsarenotevenmen-
tionedhere,itisbecauseofmylackofpersonalknowledgeofthem,aswellasthe
lackofSPACEandTIME.
ThecelebratedTheoremontheconvolutionoftwoNormallawswasconjectured
byLévywhoactuallyassumeditstruthinhisearlyworkontheinfinitelydivisible
family.ItwasHaraldCramérwhoprovedLévy’sdaringconjecture,usingproperties
ofentirefunctionincomplexanalysis.HisCambridgetract,lenttomebyProf.Pao
LuHsuintatteredcondition,wasapreciousguide.AttheBicentennialconvocation
of Princeton University in 1946, Cramér was a visiting professor who gave two
successivecoursesonmodernprobability.Iwasassignedtobehisassistant,andhe
approvedmydissertation(1947).Prof.PaoLuHsu,theninUSA,wasalsoaninvited
delegate, but he did not come (see my Preface to his Collected Papers, Springer-
Verlag, 1982 with a photo of Hsu, Cramér and Bartlett in Chapel Hill). It was on
thebicentennialoccasionthatIfirstmetFeller,Doob,AbrahamWaldamongmany
othermathematiciansknowntomebynamebefore.
OnMay19,1964Lévywrotemealetterpartofwhichannouncinghiselection
tol’Académie(inmuchdetail),theotherpartaboutMrsDoeblin’spassing.Asre-
sponseIaskedhimsoonaftertopresentaNotetotheComptesRendus(myfinalone
there).Itisabouta“pregnantintegralequation”obtainedbyconsideringalastexit
6Bytheway,itwouldbeinterestingtoshowthefollowingresultwithoutusingFouriertransform:
for any stable non-normal law with distribution function L, we have 1−L(x)∼Cx−α, for a
constant 0<α<2 as x→+∞. The problem has made some progress. The most interesting
resultisduetoLettaandPratelliwhoobtainedP(X>x)≤cx−αforanysymmetricstablelaw.
7Bull.Am.Math.Soc.(March1954).NumerousotherreferencestoLévyarementioned,including
hisprecursorMartingale.
PrefacebyKaiLaiChung ix
timefromaboundaryforMarkovchain,whichIhadspokenofinBudapest,Prague
andelsewere.AnalmosteverywheresolutioniseasybutIwasunabletoshowitev-
erywhere.HerewroteitaftertelephoningJacquesNeveufor“information”andsplit
it in two Notes which appeared in May, 1965. The open problem was first solved
by Lennart Carleson with his fine analysis, of which I wrote a reproduction in a
ChinesejournalpublishedinDalian.Lévyusedtowritetomewhilesummeringin
Menton,theborderfromVentimiglia,wheremywifeandIsojournedonce.
DuringourstayinStrasbourgin1967,whereourfive-yearoldsonwenttoFrench
school and learned to speak very well, the Lévys kindly invited us to dinner in
theirtop-floorapartmentonAve.ThéophileGautier,justabovetheresidenceofthe
novelist Mauriac, we found out later. The guests included the Chevalleys and the
Schwartzes(IfirstmetLaurentatProf.Chevalley’sofficeinPrincetonUniversity).
Monsieur Lévy offered me a little cup of Port-Salut, warning me of its parfum.
Thoseweremarvellousdays.
After the Lévy Memorial Colloque at Palaiseau in 1987, we were chez eux at
theSchwartzes,andLaurenttoldsomestoriesofhisbeau-père’speculiarmemory
of what he had or had not demonstrated the day before. Mme Schwartz was most
kindingivingmeaphotoofherfatherandoneofEmileBorel,bothreproducedin
myElementaryProbabilitybookwithsixotherluminaries.Lévyhadpassedaway
in1971.Thathappenedtobethefirstpassagetimeformyreturntomyhomeland
andtoseemymotherinHangchow,MarcoPolo’sQinsai.“TIMEMUSTHAVEA
STOP.”
In1976IwasabletodedicateapapertoLévy’smemory.8Iwasgivingacourse
on Brownian Motion (sic) with a few faculty auditors (incl. S.T. Yau now in Har-
vard). A basic theorem in Lévy’s étude profonde, proved by him using a “vertical
window,” baffled me as too hard to do in class, but I was happy when I found a
“horizontalwindow”toreplaceit,byuseofalast-exittime(Theorem1ofthemen-
tionedpaper).ThelingowasduetomyfriendanauditorfromOperationResearch
occurred.
The Brownian excursions being the groundwork for Lévy’s fantastic invention
of “local time,” I wrote with A.A. Balkema another paper entitled “Lévy’s WAY
tohislocaltime”inourSeminaronStochasticProcesses,1990(Birkhäuser).The
capitalizedWAYistakenfromLaotzeandChuangtze,pronouncedlike“TAO.”
MyacquaintancewithMauriceFréchetisbrief:Ifirstsawhimatagroupwalk
in a Japanese garden in Tokyo, in 1960. He looked very handsome and was most
cordial. Unfortunately somehow on the crowd we did not have a chance to talk.
However our first contact occurred at the time when Paris fell to the Nazis. Sud-
denlyIreceivedawholebunchofreprintssentbyhim(IwishIhadkeptthelarge
envelope,buthisreprintsarehere.)ThismustbehiskindresponsetoaNoteonan
inequalitybyE.J.GumbelIsenttoBorelforpossiblepresentationtotheAcadémie.
Fréchet also sent a postcard when my professor Pao Lu Hsu and I sent a Note on
recurrence(1946).LaterIgotagalleycopyofFréchet’sfirstvolumeonprobabili-
tiesofafinitenumberofeventsintheActualitésseries.Ihadpublishedanumberof
8Ark.Mat.14(1976).
x PrefacebyKaiLaiChung
papers on the same subject in the Annals of Mathematical Statistics (1941–1945).
Owingtothelackofcommunicationduringthewar,Fréchet’stwovolumesandmy
papersdidnotseemtorencontre.GumbelinvitedmetohishomeinBronx,New-
YorkCity,soonaftermyarrivalinUSAneartheendofNovember(1945).Wemet
againinTokyo,andwehadalastsuppertogetherattheinnwhenallotherguests
had left. Indeed, he was the first mathematician/statistician I met in America. He
wasmostkindtovisitmeinPrincetonwhereIwasagraduatestudent.Muchlater
he shared an office with a doctorate student of mine at Columbia University. This
provesthatthereisTIMErecurrenceinthismundaneworld.Ialmostforgottosay
abovethatFréchet’sGénéralitésintheBorelcollectionwasoneofmyfirstintroduc-
tion to probability after Borel’s treatise. Prof. Hsu had a copy of Fréchet’s second
volume on Markoff chains which I read. Only much later when I taught a course
on general topology I learned Fréchet’s pioneer work on that, and other kinds of
mathematics(incl.economics,IlearnedrecentlyfromProfsBruandBarbut).This
worldisconnectedinSPACEtoo.
Athome,Stanford,California
September1st,2003
