gopikrishnan 2000 aipconfproc 519 667

8/12/2019 Gopikrishnan 2000 Aipconfproc 519 667

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Financial Time Series:APhysics PerspectiveParameswaranGopikrishnan, Vasiliki Plerou, Luis.A. N.Amaral,

Bernd Rosenow,aHEugene Stanley*Centerfor Polymer Studies1 and Department ofPhysics,

Boston University, Boston MA 02215, USA.

Abstract. Physicists in thelastfew years havestartedapplying concepts and methodsof statistical physics to understand economic phenomena. The word econophysicsissometimes usedo refero this work.Oreasono this interesth fact thatEconomic systems sucha financial marketsa examplesocomplex interacting sys-temsowhichahuge amountodataexistapossiblethateconomic problemsviewed froma different perspective might yieldnwresults. This article reviewshresults of a fewrecent phenomenological studies focussedon understanding the dis-tinctive statisticalpropertiesofinancial time series.Wdiscuss three recent resultsT probability distributiono stockprice fluctuations:Stock price fluctuationsoccur in allmagnitudes, inanalogyto earthquakes from tiny fluctuationsto verydrastic events, such as market crashes, eg., the crash of October 19th 1987, sometimesreferred to as Black Monday . The distribution of price fluctuations decays with apower-lawtail welloutside the Levystableregime and describes fluctuationsthatdif-fer by as muchas 8ordersofmagnitude. In addition, this distribution preserves itsfunctional form for fluctuationson time scales that differ by 3orders ofmagnitude,from 1 min up to approximately 10 days, (ii) Correlations in financial time series:While price fluctuations themselves have rapidly decaying correlations, the magnitudeof fluctuations measuredbyeither theabsolute valueor thesquareof thepricefluctu-ationshcorrelations thatdecayaapower-lawapersistoseveral months, (iii)Correlations among different companies:T third result bearsoh applicationorandom matrix theory to understand the correlations among price fluctuations of anytwo different stocks. Prom a study of the eigenvaluestatisticsof the cross-correlationmatrix constructed fromprice fluctuationsoh leading 1000 stocks,wfind thathlargest 5-10% of the eigenvalues and the corresponding eigenvectors show systematicdeviationsfromthe predictions for a random matrix, whereas the rest of the eigenvaluesconformo random matrix behaviorsuggestingthat these 5-10%oheigenvaluescontain system-specific information about correlated behavior.

The Center forPolymer Studiesissupportedby theNational Science Foundation.

CP519,Statistical Physics,edited by M. Tokuyama and H. E. Stanley2000American Institute of Physics l-56396-940-8/00/$17.00667

Downloaded 13 Jan 2009 to 128.197.42.82. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/proceedings/cpcr.jsp
http://dx.doi.org/10.1063/1.1291641


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INTRODUCTIONWhatsh question?

Statistical physics describes physical systems consistingoa large numberointeracting units. On one hand, economic systems such as financial markets aresimilar to physical systems in that they are comprised of a large number of in-teracting agents . On the other hand, they are quite different and much morecomplicated because economic agentsa thinking unitsa they interactncomplicated ways not yet quantified. Indeed, most physics approaches to financeview financial markets as a complex evolvingsystem. Are there any analogs foreconomicsof physical laws thatdescribe complex physical systems? Indeed, theevolutionovarious financial time seriesa randomjustah motionoaparticleundergoingBrownianmotionaaasuch governedbprobabilisticlaws. Specifically, webegin by asking what the statistical features that describeth evolutionofinancial time seriesa [1-10].

Why do we care?Apart from its practical importance and its importance in mordern economics,th scientific interestnstudyingfinancial marketsstems fromh fact that thereisawealthodataavailableofinancialmarketswhich makesarguablyho

complex system most amenable to quantification and ultimately scientific under-standing. In addition, it is also possible that the dynamics underlying financialmarketsa universal a exemplifiedn several studies which have notedhstatisticalsimilarityohpropertiesoobservablesacross quitedifferent markets.Nodoubt,economistshavestudied these systemsolong.Omight thinkwhata physicists can contribute to such a field. In general, thereare twopossibilities:(i) In Economics, one often starts with a model and tests what the data can sayabouthmodel.Ownwhichphysicistsccontributeothisfieldostartinh spiritoexperimental physics whereotriesouncoverh empiricall wswhicholatermodels (ii)nalmostaomodern physicstheoriesstochasticityorandombehavioringrained, froma simple random walkoquantum mechanics.Many sophisticated methods were developedo understandh dynamicsosys-temswith manyconstituent elementswhich behave in anon-deterministicfashion.Hence, in a general sense, it ispossible that the methodsand concepts of modernphysicsccontributenthisrealm,ahbeen shownbseveralphysicistsnhfewrecent years. So, thequestion really is: Canmethodsand conceptsdeveloped inphysicsbusefulneconomics? Could economic problems approachednadifferentpoint ofview,yield newinsights? Thus, fo rmanyphysicists, studying the economymeans studying a wealth ofdataon a stronglyfluctuating complex system. Indeed,

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physicistsn increasing numbers (Fig.1a finding problems posedbeconomicssufficiently challengingto engage their attention [11-27].

8 00ort 60

40do 20

LLJ

Cond-matPhysicaA

01994 1996 1998 2000FIGURE 1. The number of articles on econophysics that have appeared on the cond-matpreprintserver and in the journalPhysicaA as ofOctober 1999, obtained from the Los AlamosWebserver and the Sciencecitation index respectively using the keywords stock, market, price,finance, financial, an d options

What do we do?

Recent studies attempto uncovera explainh statisticalpropertiesonancialtime seriessuchas stock prices,stock marketindices or currency exchangerates. The interactions between the different agents comprising financial marketsgeneratemanyobservablessuchas thetransactionprice,the sharevolumetraded,the trading frequency, and the values ofmarket indices. Recent empiricalstudiesare based on the analysis of price fluctuations. This talk reviews recent resultson (a) the distribution of stock price fluctuations and its scaling properties, (b )time-correlationsnfinancialtime series,acorrelations amonghprice fluc-tuationsofdifferent stocks. Spacelimitationsrestrictus to focusingmainlyon ourgroup'swork;a morebalanced accountcan befound in tworecentbooks [6,5],otherarticlesintheseproceedings,and twoother recentinternational conferences [8-10].Recent work inthisfield alsofocuses onapplicationssuchas riskcontrol,derivativepricing,a portfolio selection [29], which shallnbdiscussednthistalk.Tinterested reader should consult, forexample, Refs. [6,5,30].

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DISTRIBUTION OFPRICE FLUCTUATIONST recent availabilityo high frequency data allowsoo study economictimeseriesoawiderangeotimescalesvaryingfromsecondsuoawmonths.Fexample,orecent work [19,41,42] involveshanalysisohS&5index,a indexohNwYork Stock Exchange that consistso5companies repre-sentativeohUeconomy.a market-value (stock price times numbero

shares outstanding)weighted index, with eachstock'sweight in the indexpropor-tionateomarketvalue [19].TS&P5indexoohmostwidelyusedbenchmarksoU.S. equity performance.Wanalyzed high frequency dataoh13-yearperiod 1984-96with arecordingfrequencyof oneminuteorshorterand thedaily records for the 35-yearperiod 1962-96 Fig. 2a).

The S&P 500 index Z(t) from 1962-96 has an overallupward driftinterruptedby drastic events such as the market crash of October 19, 1987 Fig. 2 a). Oneanalyzesh differencen logarithmoh index, often calledh return G(t)=logeZ+At) logeZ i), whereAh time scale investigated Fig. 2b).Oonlycounts the numberof minutes during theopeninghours of the stockmarket. Itisapparent from Fig.2that whenoanalyzes returnsoshort timescales, largeevents are much more likely to occur, in contrast to a sequence ofGaussian dis-tributed randomnumbers of the same variance Fig. 2d). As oneanalyzesreturnson largertimescales, thisdifferenceapparently much less pronounced Fig. 2c).In orderounderstand thisprocess,ostartsbanalyzinghprobabilitydistri-butionoreturnsoa given timescaleA whichno study, varies from1mn

8Q_o300 2102 A /H SJ

A/

.

/:/fGftNnS t+At)/S t) -1 (a)

i i

IO

0 03 150^ 0

-150

bS51mndata),^Wl[44Y^

(c) S P 500 (monthlydata);P(d)Gaussiannoise

tyf ^^

Year TimeFIGURE 2. (a) The daily records of the S&P 500index for the 35-year period 1962-96 on alinear-log scale.Note thelargejumpwhichoccured during themarket crashofOctober19, 1987.Sequenceob1mn returnsa1month returnsohS&5 index, normalizedounit variance,d Sequenceo i.i.d. Gaussian random variables with unit variance, whichwproposedbBachelieraa modelostock returns [1].Fa3panels, therea8eventsi.e., in panel (b) 850 minutes and inpanel (c) 850 months. Notethat, incontrast to (b) and c) , thereare no large events in d) .

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up to a fewmonths.T natureoh distributionoprice fluctuationsn financial time seriesalongstandingopen problem infinance whichdatesback to the turnof the century.In 1900, Bachelierproposed the first modelfor the stochasticprocess of returnsan uncorrelated random walk with independent, identically Gaussian distributed(i.i.d) randomvariables [1]. Thismodelisnaturalif oneconsidersthereturn overatime scaleAt to be theresultof alargenumber ofindependent shocks ,whichthenleadby thecentral limittheorem to aGaussian distributionof returns [1]. However,empirical studies [4,19,20] show thath distributionoreturnsh pronouncedtails in striking contrast to that of a Gaussian. Despite this empirical fact, theGaussian assumption for the distribution ofreturns is widely used intheoreticalfinance because of the simplifications it provides in analytical calculation; indeed,it is one of theassumptionsusedin theclassicBlack-Scholesoptionpricingformula[31].In his pioneeringanalysisofcottonprices,Mandelbrot observedthat inadditiontobeingnon-Gaussian,hprocessoreturnsshowsanother interesting property: time scaling thath distributionso returnso various choicesoAranging from 1 day up to 1 month have similar functional forms [4]. Motivatedby (i)pronounced tails, and (ii) a stablefunctional form fo rdifferent timescales,Mandelbrot [4]proposed that the distribution ofreturns isconsistent with a Levystabledistribution [2,3].

Conclusive results on the distribution ofreturns are difficult to obtain, and re-quirea large amountodatao studyh rare events that give riseoh tails.More recently, the availability of high frequency data on financial market indices,ah adventoimproved computingcapabilities,h facilitatedh probingoth asymptotic behavioroh distribution.F example, Mantegnaa Stan-ley [19] analyzed approximately1million recordsohS&5 index. Theyreport thath centralpartoh distributionoS&5returnsappearsobwellbaLevy distribution,bh asymptoticbehavioroh distributionoreturns shows faster decay thanpredictedbaLevy distribution. Hence, Ref. [19]proposedatruncated LevydistributionaLevydistributionnhcentralpartfol-lowedbaapproximately exponential truncationasamodelohdistributionofreturns. The exponentialtruncationensures the existence of a finite second mo-ment,ahencehtruncatedLevydistributionnastabledistribution [33,34].The truncated Levy processwith i.i.d. randomvariableshas slow convergence toGaussianbehaviordohLevydistributionnh center, whichcouldexplainth observed time scalingoaconsiderablerangeotime scales [19].

Recent studies [35,36]oconsiderably larger time series using larger databasesshowquitedifferent asymptoticbehaviorfor thedistributiono freturns. Ourrecentwork [35] analyzed three different data bases covering securities fromh threemajorUstock markets.n total,wanalyzed approximately4millionrecordsostockprices sampleda5mnintervalsoh 1000leadingUstocksoh2-yearperiod 1994-95 and 35 milliondaily records for 16,000 US stocks for the 35-yearperiod 1962-96. Westudy theprobabilitydistributionofreturns (Fig. 3(a,b,c)) for

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individual stocks overa time intervalA whereA varies approximately overafactor of 104from 1 min up to more than 1month. Wealsoconduct a parallelstudyof the S&P 500 index.Our keyfindingisthat the cumulativedistributionofreturnsforbothindividualcompanies (Fig.3ahS&5index (Fig.3cbwell describedba

power lawasymptoticbehavior,characterizedby anexponent a 3, welloutsideth stable Levy regime0


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althoughna stable distribution,retainsfunctional formotime scalesuoapproximately1daysoindividualstocksa approximately4daysohS&500 index, Fig. 3b. For larger time scales our results are consistent with break-downoscalingbehavior, i.e., convergenceo Gaussian [35]. Similar resultshavealso been found fo rcurrency exchangedata [36].

CORRELATIONSINFINANCIAL TIMESERIESIn additionoh probability distribution,a aspecto equal importanceo

the characterizationof anystochasticprocess is the quantification ofcorrelations.Studiesof the autocorrelation functionof the returns show exponentialdecay withcharacteristic decay timesoonly4mn [37] consistent withh efficient markethypothesis [38]. Thisparadoxical,onhprevious section,whaveseen thatth distributiono returns,n spiteo beinga non-stable distribution, preservesits shapefor a wide range ofAt Hence, there has to be somesort of correlationsor dependencies that preventh central limit theoremo take over soonerapreserve the scaling behavior.Indeed, lack oflinear correlation does not implyindependent returns, sincetheremayexist higher-ordercorrelations. Recently,Liu and hiscollaboratorsfound thatth amplitudeoh returns,h absolute valueoh squareclosely relatedto what is referred to in economics as the volatility [39] shows long-rangecorrelations [22,24,40-43] with persistence [44] up to several months, Fig. 4(a,b).They analyzed the correlations in the absolutevalue of the returns [41,42]of theS&5indexusingtraditionalcorrelationfunction estimates,powerspectrumathrecently-developeddetrendedfluctuationanalysis(DFA).Ahthreemethodsshowh existenceo power-law correlations witha cross-overa approximately15days.FhS&5index,DAestimatesoh exponentscharacterizingth powerawcorrelationsao= 0.66o short time scales smaller than 15daysa a =0.93olongertime scalesuoayear, Fig.4Findividualcompanies, the same methods yield a\ 0.60 and a2 0.74, respectively. Th epowerspectrumgivesconsistentestimatesohwpower-lawexponents,Fig.4

What is Volatility?The long memoryin the amplitude ofreturnssuggeststhat it is useful to define

a subsidiary process, referred to as the volatility. Volatility of a certain stockmeasures howmuch it islikelyto fluctuate. It canalsoberelatedto theamountofinformation arrivingaa time.T volatilitycbestimatedoexamplebthe localaverageof theabsolute valuesor the squares of the returns.Intheir recent work on the statisticalproperties ofvolatility Liu et. al. for theS&5index,Fig.5 They showthathvolatility correlationsshowasymptoticI behavior [41-43]. Usingh same data basesa above,Luahcollabo-rators also study the cumulativedistribution ofvolatility [41,43] and find that it

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1 0 .

10-3

tone day (a)Shuffled data 5=0

P2=0.90

10 10 10 11f[min]

103

FIGURE 4 Plotoh power spectrum S(f)abh detrended fluctuation analysisF i)oh absolute valuesoreturns g ( t ) , after detrendingh dailypattern [41,42]withhsamplingtime interval At = 1min. Thelines show the best power lawfits R valuesarebetterthan 0.99)abovea belowh crossover frequencyo/ x= (1/570)min 1naohcrossover time,tx =600minin(b).The trianglesshowthepowerspectrumand DFA results forth control ,i.e.,shuffled data.

isconsistent with apow er-lawasymptoticbehavior, characterized by an exponent// 3, just thesameasthatfor the distribution of returns. Forindividualcompa-nies also,ofindsa similar powerawasymptoticbehavior [42].naddition,also found that thevolatility distribution scalesfor arangeof time intervalsjustas the distributionofreturns.

CORRELATIONS AMONG DIFFERENTUNITSRecently, the problem of understanding the correlations among the returns of

different stockshbeenaddressedbapplyingmethodsorandom matrixtheoryto the cross-correlation matrix [47-49]. Aside from scientific interest, the studyofcorrelations between the returns ofdifferent stocks isalso ofpracticalrelevancein quantifying the risk of a given portfolio [29]. Consider, forexample, the equal-timecorrelationostock returnsoa given pairocompanies. Sinceh marketconditionsmnbstationary,ahhistoricalrecordsafinite,nclearifa measuredcorrelationof returns of twostocksisjustdue to noise orgenuinelyarises fromh interactions amonghwcompanies. Moreover, unlike mostphysical systems, theren algorithm o calculateh interaction strengthbetweenwcompaniestherefor,say,wspinsnamagnet).Tproblemthatalthough everypairocompanies shouldinteract eitherdirectlyoindirectly,the precisenature ofinteraction is unknown.In some ways, the problem ofinterpreting the correlations between individual

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1.0 -84 85 86 87 88 89 90 91 92 93 94 95 96 97Time[yr]

FIGURE51-min recordsohS&5indexoh13-yearperiod 1984-96, containingapproximately1.2 millionrecords, (b)VolatilityVr t) estimatedusingwith anaveragingwindowof T =1 month (8190min)and sampling timeintervalA t = 30 min of the S&P 5 00 index for theentire 13-year period 1984-96.Thighlightedblock showspossible precursors ohOct.8crashpostulatednRef.[15].

stock-returnsisreminiscentof the difficultiesexperiencedbyphysicistsin the fifties,ininterpretingthespectra ofcomplex nuclei. Largeamountsofspectroscopicdataon the energy levelswere becomingavailableb ut were toocomplexto beexplainedby modelcalculationsbecauseh exact natureoh interactionswereunknown.Random matrixtheory (RMT) was developed in this context, to deal with thestatistics of energy levels of complex quantum systems [50-52]. With the mini-malassumptionof a random Hamiltonian, given by areal symmetric matrixwithindependent random elements, a series of remarkablepredictions were made andsuccessfully testedoh spectraocomplex nuclei [50].RMpredictions repre-sent an average over all possibleinteractions [51]. Deviations from the universalpredictionsoRMidentify system-specific, non-random propertiesoh systemunderconsideration,providing cluesabout the underlyinginteractions [52]

Recently, Plerou and her collaborators analyzed the cross-correlation matrix C=C=(GiGj) (Gi)(Gj)/(7i(jjoh returnsa 30-minuteintervalsoh largest1000USstocksfor the 2-year period 1994-95.They analyzethestatisticalproper-tiesof C by applying techniques ofrandommatrixtheory (RMT) [47,48]. First,theytestheigenvaluestatisticsohcross-correlation matrixouniversalpropertiesofreal symmetric random matrices such as the Wigner surmise for the eigenvaluespacingdistributionandeigenvaluecorrelations.Remarkably, they find that eigen-

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valuestatisticsoh correlationmatrixagree wellwithh universal predictionsof random matrix theory for real symmetric random matrices, in contrast to ournaiveexpectationsfor a stronglyinteractingsystem.

Deviations fromRMpredictions represent genuine correlations.n orderoinvestigate deviations,wecomputethe distributionof the eigenvalues of the C andcomparewithhprediction [47]ouncorrelatedtimeseries [53].Wfind thathstatisticsof all but a few of the largest eigenvaluesin thespectrumof Cagree withthe predictionsofrandom matr ix theory, but there are deviationsfor a few of thelargest eigenvalues [47,48].Tdeviationsoh largestweigenvalues fromhrandommatrixresultaalsofoundwhenoanalyzeshdistributionoeigenvec-to components. Specifically,h largest eigenvaluewhichdeviatessignificantly2timeslargerthanrandom matrixbound) has almost allcomponentsparticipatingequally and thusrepresents the correlationsthat pervade through the entiremar-ket. Thisresultn agreement withh resultsoLalouxa collaborators [47]forh eigenvaluedistributionoCoadaily time scale.

CONCLUSIONThe empiricalresultsshown above clearlybeckon explanation. For example, infirst two sections, we have looked mainly at twoempirical results: (i) the distri-

butiono fluctuations, which showsa powerawbehavior welloutsideh stableLevy regime,ay preservesshapescalesfora rangeo time scalesa(ii)h longrange correlationsnh amplitudeoprice fluctuations.Hwahtworelated ?

Previous explanations of scaling relied on Levy stable [4] and exponentially-truncated Levy processes [5,19]. How ever, the empirical data that we analyzean consistent with eithero thesewprocesses.n ordero confirm thatthe scaling is not due to a stable distribution, one can randomize the time seriesof1mn returns, thereby creatinganwtime series which contains statistically-independent returns.Baddingunconsecutivereturnsohshuffled series,ocan constructh nmin returns. Bothh distributiona moments showarapid convergenceo Gaussianbehavior with increasingn showingthath timedependencies, specifically volatility correlations are intimately connected to theobserved scalingbehavior [35].

Usingh statisticalproperties summarized above,cwattemptodeduceastatistical descriptionohprocesswhichgives riseo thisoutput?F example,thestandardARCHmodel[28,45]reproducesthepower-lawdistributionofreturns;however it assumesfinite memory onpast events and henceis not consistent withlong-rangecorrelationsinvolatility.On theotherhand,thedistributionofvolatil-ity and that of returns which have similar asymptoticbehavior, however supportthe central ARCH hypothesis that g(t) ev t), where e is an i.i.d. Gaussianrandom variable independent of the volatility v( t ) , and g(t) denotes the returns.A consistent statistical description may involve extending the traditional ARCH

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modelo include long-rangevolatilitycorrelations [46].Amore fundamental questionwouldbounderstandhaboveresultsstartingfrom amicroscopicsetting. Researchershavealso studiedmicroscopic modelsthatmight give rise to the empirically observedstatistical propertiesofreturns [6,11].For example, Lux and Marchesi [11]recently simulated a microscopic model offinancial markets withwtypesotraders,what they referoa 'fundamentalist'and 'noise'traders.Their results reproduce the power-law tailfor the distributionof returnsand alsothe longrange correlationsin volatility.In thelast section,wefound evidencefordifferent modeso fcorrelationsbetweendifferent companies. For example, the largest eigenvalue of the cross-correlationmatrix showed correlationsthat pervadeh entire market. Couldb thathabove observed scalingpropertiesare related to how correlationspropogatefromoneunitohothersuchaoccurncritical phenomena? Researchers have studiedeconomic data fromh physicsperspectiveoa complex system with each unitdependingoh other. Specifically,h possibilitythatah companiesnagiven economymight interact, moreorless,likeaspin glass. In aspin glass, eachspininteractswitheveryotherspinbutn withhsamecouplingan evenwithh same sign.F example,h stock priceoa given business firmAdecreasebe.g.,1thiswillhavea impactnheconomy. Someothesewillbefavorablefirm B,whichcompeteswith A, mayexperienceanincreaseinmarketshare. Otherswillbnegativeserviceindustriesthatprovide personal servicesofirmAemployeesmexperienceadrop-offnsalesaemployeesalarieswillsurelydecline.There mustbpositivea negative correlationsoalmostaeconomicchange. Can weview the economyas acomplicated spin glass?To approch this problem, M. H. R. Stanley and M. A.Salinger first locatedandsecuredadatabasecalledCOMPUSTATthatlistshannual salesoevery firminthe UnitedStates. With this information,M. H. R.Stanleyand co-workerscal-culatedhistogramsohwfirm sizes change fromoyearoh next [54].Theyfind that the distribution of growth rates of firm sales has the same functionalform regardlessoindustryomarket capitalization. Moreover,h widthothesedistributions decrease with increasing salesaa power-law witha exponentaproximately1/6. Recently,similarstatisticalpropertieswerefoundohGocountries [55]aouniversity research fundings [56].Hence,n impossibleto imagine that there are some very general principles of complexorganizationsa work here,because similar empiricallawsappearoholdodataoa rangeosystemsthatat firstsight might not seemto be socloselyrelated. Buldyrev modelsthis firm structureas anapproximateCayleytree, in which eachsubunito f a firmreactsto its directivesfromabove with acertain probability distribution [57].Morerecently,Amaralea[58] haveproposedamicroscopicmodelthat reproducesboththe exponent and the distribution function. Takayasu and Okuyama [59]extendedth empiricalresultsoawiderangeocountries.We concludeb thankingao collaboratorsa colleagues from whomwlearned a great deal. These include the researchers and faculty visitors to ourresearch group with whom we have enjoyed the pleasure of scientific collabora-

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