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Record Values and Maxima

Author(s): Sidney I. ResnickSource: The Annals of Probability, Vol. 1, No. 4 (Aug., 1973), pp. 650-662Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2959435.

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The Annals of Probability1973,Vol. 1, No. 4, 650-662

RECORD VALUES AND MAXIMABY SIDNEY I. RESNICK

StanfordUniversity{X , n_ 1} arei.i.d. random ariableswith ontinuous fF(x). Xj isa recordvalue of this sequence ifXj > max {X1,* ,Xj-l}. We comparethebehavior f the equence frecord alues XLn} with hat fthe amplemaxima {Ma} = {max XI, * **,X )}I. Conditions for the relative stability(a.s. and i.p.) of XL } aregivenand in eachcase these onditions mplythe relative tability f {Ma}. In particular egular ariation f R(x) _-log (1 - F(x)) is an easilyverifiedonditionwhich nsures .s. stabilityof XL }, {M,} and {E=1 Mj}. Concerningimitaws,XL, may onvergein distribution ithoutM,} having limitdistributionnd vice versa.Suitable differentiabilityonditions n F(x) insure hat bothsequenceshavea limit istribution.

1. Introductionndpreliminaries. et {X,,,n > 1}be a sequenceof ndependent,identicallydistributedi.i.d.) random variables withcommondistribution (.).X, is a record alue of this equenceiffXj > max {X1,*.., Xj_1}. ByconventionX1 s a record value. The indices at whichrecordvalues occur are givenby therandomvariables {L,,,n > 01definedby Lo = 1, L_= min ]j j > L,-,, Xj >XL_-1}. We assumethroughout hatF(.) is continuous n order to be able touse nonparametric echniques n our analysis.The emphasisof thispaper is on therelationships f the XL,} sequenceto thesequenceofmaxima M } whereM,,= max {X1, ***, X }. Theserelationshipsreweakerthanexpected,even though XL,,} is an embeddedsubsequenceof {M,}.A sequence of random variables {Z.} is relativelytable henceforthuststable)if there exist normalizing constants {B.} such that Zn/Bn + 1 as n -* oo. If theconvergence s withprobability1 then ZJ} is almost urely a.s.) stablewhile ftheconvergence s in measure we say {Z.} is stable nprobabilityi.p.).

In thefollowing ectionswe determinetability onditions or XLI, {MI,}and{E ,, MJ and show {XL I stable implies M,} stable but not the reverse. Theapproach s toreducequestions bout extremes o questionsabout sumsof i.i.d.randomvariables. In the last section we discuss the relationship etweentheexistenceof limit aws for AMn}nd existenceof limit aws for XL}.A central role will be played by R(x) = -log (I - F(x)) and its inverseR-1(x) = inf y R(y) > x}. BothR and R-1 are non-decreasing; maps [-00,oo] onto[0, o0]. Ify = R(x) thenwehave R-1(y-) < x < R-1(y), but for con-veniencewe shall simplywrite x = R-'(y). The asymptoticswe deal withwillbe unaffectedy thisconvention.ReceivedFebruary 4, 1972;revised anuary , 1973.AMS 1970 ubject lassifications.rimary0F05.Key wordsndphrases.Recordvalues, maxima, xtreme alues, regular ariation, imitingdistributions,elative tability. 650

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RECORD VALUES AND MAXIMA 651This significantropertyfR(x) is that f a randomvariableX has distributionF(x), thenR(X) is exponentially istributed: [R(X) ? x] = 1 - e-x for x ? 0.The followingbasic lemma was presentedn [16]:LEMMA 1. {X., n ? i1 is an i.i.d. sequencewithP[X,, ? x] = 1, ezx,x ? 0.ThenXL XL =(X, XJ1 Z Yj where Yj,] ? O1is an i.i.d.sequence istributedccording o the ameexponential istribution.This lemma and the fact that R is non-decreasingnd continuous ead to thefollowing epresentations:(1) ~~~R(XL z~v=0J XL >R1E% Y')

wherefY6,] ? 0} are i. . d. exponentially istributed andomvariables.It is convenient o define the right nd x0of a distribution unctionF(x) asX0= sup x F(x) < I}.To every distributionfunctionF(x) correspondsthe associated distributionF a) (x) defined to be that distributionwhose R-function s Ri(x). Formally:1 - F~ )(X) =_ e-R(x) for -~ ?o x_'Thetheory f limitaws -forecordvalues is closelyrelated to extremevaluetheory see [16]). The following s pertinent: Let {X., n ?,,> ) be i.i.d. withcommon distribution (x). If there exist normalizingconstantsa, > 0,

such that(2) P[a j1(M, - /) x] =- F11a x + j)- G(x)whereG is nondegenerate, henG belongsto thetypeof one of the three xtremevaluedistributions2], [10]:

A(x) exp{-e-x} -co < x < 00(Pjx) 0 if x 0wherea is a positiveconstant. Abbreviate (2) by F c DM(G) which indicatesthat normalized maximafromF converge n distribution o G.Similarly XL}n has limiting ecordvalue distribution (.) ifnormalizing on-stants . > 0, b. exist such that(3) P[XL ? a x + bn] H(x)where H is nondegenerate.Write R c DR(H) to indicate weak convergence fthe recordvalues to H. In [16] it was proved that the limitingrecord valuedistributions re of the formN( - log (- logG(x))) where N is the standardnormal distribution nd G is an extremevalue distribution.FurtherR(x) GDR(N(- log (-log G(x))) iff he associated distribution ~a) x) c DM(G(x)).The equivalence betweenrecord value behavior of a distributionnd extreme


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652 SIDNEY I. RESNICKvalue behavior of the associated distribution s valid also for stability .p. butbreaks down for a.s. stability.

2. Stability n probability. n this and the following ection we assume F hasinfinite ight nd. In thecontrary ase when x0< oo we have Mn x0 .s. andXLn XO.s.It is wellknown [2], [10]) that M.} is stable .p. iff - F(x) is rapidly arying:(4) lim, (1 - F(tx))/(1 - F(x)) = 0for all t > 1. In this case it is always true thatM./R-'(logn)Up 1.

THEOREM 1. Thefollowing re equivalent:(i) There xistB. > 0 such thatXL,/B. --Up 1.(ii) XL /R-'(n) >p1.(iii) lim, {R(tx) - R(x)}/Ri(tx) = oo for all t > 1.(iv) lima. {R(tx) - R(x)}/Ri(x) = oo for all t > 1.

PROOF. The equivalence of (i), (ii), (iii) was proven n [16], Theorem2.1 andobviously iii) implies iv). Given (iv) suppose (iii) does not hold. Then thereexistt > 1, x -0 oo and c < oo such that im, . {R(tx,) - R(x.)}/RI(tx.) = c.This with iv) entails im,,. R(x,)/R(tx,)= 0 so that{R(tx.) - R(x,)}/Ri(tx,) = {Ri(tx,) - Ri(xj)}{Ri(txj) + Ri(xj)}/Ri(txn)

= (1 - o(I)){R1(txn) + RI(x)} -> 00as n - 0oo whichgivesa contradiction. This completestheproof.

THEOREM2. {XL } is stable .p. iffmaxima f .i.d. random ariables istributedaccording o the ssociateddistribution'a)(x) are stable .p.; i.e., iff1 - F(a)(x) israpidly arying.PROOF. From (iii) ofTheorem1 we have:

00 1. This is equivalent to F(a, satisfying 4). The converseis almostthesame.

COROLLARY1. {XL} is stable .p. iffR '((log x)2) is slowlyvaryingffR-'(x +cxi) - R-'(x) as x -+ oo for all real c.PROOF. A distributionH satisfies 5) iffH-1( - x-1) is slowly varying [2]),SO {XL.} is stable iff (a)(x) satisfies 4) iffRi)-1(log x) = R-'((log x)2) is slowlyvarying. The second statement f the corollaryfollows by a change of variable.THEOREM 3. XL /R-'(n) -U>p1 implies M./R-1(log n) -->p 1, but the converse isnot true.


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RECORD VALUES AND MAXIMA 653PROOF. If {XL } is stable .p. thenTheorem1 iv) holds,which entailsR(tx) -R(x) -+ oo as x -0> oo for all t> 1. This is equivalentto therapidvariationofl -F(x).As a counterexample o the converseconsiderR(x) = (log x)2, x > 1. ThenR-1(logy) = exp{(logy)i} is slowly varyingso that {Mj} is stable i.p., butR-'((logy)2) = y is not slowly varying. Hence {XLq,,} fails to be stable i.p.3. Almost ure tability. n discussing .s. stability f {XL }, it is clear fromTheorem1 thatwe needconsiderno othernormalizing onstants xcept R-'(n)}.For what follows it is convenientto set Zn = (Z>% Yj - n)/(2nog logn)iwhere {Yj, > 0) are the i i.d. exponentiallydistributed andomvariables ofrepresentation1). It follows by the Law of the IteratedLogarithm 12] that

almost surely:(5) limSUP,,,,. n' I , lim nfer,-.n, I1THEOREM 4. A sufficientonditionorlima O. L /R-'(n) = 1 a.s. is:

(6) For all real t: lim, 00R-'(s + t(s log log s)i)/R-'(s) = 1 .REMARKS CONCERNING CONDITION (6). Note first hat theconvergence n (6)is uniform n finite intervals. Bythe nversion echniquedescribed n [6] (seealso [2], [4]), (6) is fully quivalentto the moreeasilyverified ondition:

(7) Vx > 1: limt,. {R(tx) - R(t)}/(2R(t) og log R(t))i = ooIt is clear that (7) impliesTheorem 1 iii), so either 6) or (7) is sufficient orXL /R-'(n) -np1. Note if R-1 is regularly aryingwith xponent , 0 < a < oo(equivalentto R regularly aryingwithexponent -' via [2]), then 6) or (7) issatisfied.PROOF OF THEOREM 4. Keepingin mind the uniform onvergence n (6), wehave: lim,_O0 L /R-1(n)= 0R-'(E%_ Y.)/R-'(n)

= limD00R-1(n+ Zn(2n og log n)i)/R-1(n)_ 1 a.s.where thefirst qualityfollowsby (1), the second by the definition f Zn andthelast by (5).

REMARKS. (i) Condition (6) is sufficient ut not necessaryfora.s. stability.A counterexamplehas been constructedby A. A. Balkema and will appearelsewhere.(ii) Almostsurestabilitywill nothold ifforall x:limt.00 -1(t+ x(2t og log t)i)/R-1(t) eex 0 < c ? 00 .See [6].(iii) Added information bout how close (6) is to a necessaryand sufficientcondition sprovidedbythefollowing easoning:For anycontinuousdistribution

XL IR-'(n) = R-'(n + Z (2n log log n)i)/R-1(n) o that imsup,,,. XL IR-1(n) I


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654 SIDNEY I. RESNICKa.s. and lim nfnO. LI/R-'(n) < 1 a.s. Hence stability s equivalent to

0 = P[XLn > (1 + s)R1(n) io.] = P[XL_< (1 - s)R'(n) i.o.],forall s > 0, which from 1) is the same as:(8) P[E>0 Y3 > R((l + s)R-1(n+ 1)) i.o.]

= P[L 0oYj < R((l - s)R-1(n)) i.o.] =We wishto applyFeller's generalform f the law of theiterated ogarithm7],but to do this we must suppose the followingmonotone convergences:(9) lim, {R(tx) - R(x)}/Ri(tx) o(10) lim {R(tx) - R(x)}/R1(x)= o0for all t > 1 (cf. Theorem3). Supposing 9) and (10), (8) is equivalent to theconvergenceof two serieswhich in turncan be shownequivalentto thecon-vergenceof two integrals. The convergenceof the integralcorresponding olim nfnO.XL /R-1(n) 1necessitates onvergence f the ntegral orrespondingto limsup,,,. XLn/R-'(n) 1. The result s that under 9) and (10) a.s. stabilityof {XLJ is equivalentto( 1) Ve > 0 S0 R(y) - R((l - s)y)(11) ~~~~~~RI(y)x exp 1 ( R(y) - R((1- s)y) ) d log R(y) < oo(2 RP(y) o ~) 1 a.s. is (6).

The possibility fother imitpointsbesides1forXL /R-'(n)} sdiscussed n 6].For what followswe need thefollowing esult 1], [17]: Suppose F(x) < 1 forall x. Then there xistnormalizing onstraints e, n > 1 such that M./b. -+ 1a.s. iff(12) For all 0 < s < 1: S1 - F((1 - e)x))-1 dF(x) < oo.In this case b. ~ R-1(logn), n oo.

THEOREM 5. Condition6) is also sufficientorlim,_. Mn/R'-(logn) = 1 a.s.PROOF. Let p(n) be thenumberof recordvalues in {X1, **,X.j so thatM. =

XL,(,,). An iteratedogarithm heoremholds forpa(n)see [14]) and thisgivestherepresentation(n) = log n+ Zn*(2 og n og log log n)i where imSup,-. Z.* = 1,lim nf,,. Z* =- 1, a.s. HenceMA/R-1(log ) = {XL (fn)/RR-(p(n))}{R-1(p(n))/R-'(logn)}


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RECORD VALUES AND MAXIMA 655The first actor onverges o 1 a.s. byTheorem4, and thesecond factorequalsR-1(logn + Zn*(2 log n og log logn)i)/R-1(log ) whichconverges o 1 via (6).

REMARK. When (9) and (10) hold, we have byProposition1 thatlim n00 XL /R-1(n) 1 a.s. lim,-O.MJ/R-1(log) = 1 a.s.

EXAMPLES. (i) Let R(x) = (log x)2,x > e. In Theorem 3 we showedthatXLcomingfrom hisR-function re not stable i.p. Hence {XL} is not a.s. stable.However, (12) reduces to a convergent amma integral nd hence {Mj is a.s.stable.(ii) Let N(x) be the standard normal distribution with density n(x)-(2w)-ie-X2/2.henas iswellknown: 1 - N(x) n(x)/x s x -+ oo,whichentailsR(x) = -log (1 - F(x)) --log n(x) + logx -x2 s x -- oo so thatR is regu-larly varyingexponent2. Hence (5) is satisfied, nd since R-1(x) , (2x)#asX-o 00, we have a.s.:

limn0 XL_/(2n)i 1 limn. Mn/(2ogn)i = 1IThe close relationship etweenrecordvalues fromthe distribution (x) andmaxima fromthe associated distribution (a)(x) whichwas shownto hold forlimit aws and stability,now breaks down:THEOREM 6. Undercondition9) ifmaxima rom F(a)(x) = 1-exp{-Ri(x)}are a.s. stable, thenrecordvaluesfrom F(x) = 1 - exp{-R(x)} are a.s. stable.The conversesfalse.PROOF. According o 12), a.s. stability f maximafrom (a)(x) isequivalentto

Ve, 0 exp{-(Ri(y) - Ri((1 - s)y))} dRi(y) < o0oTo show {XL } a.s. stable,we verify7). Note, however,that a simple proofbycontradiction rgument similarto the one used in Theorem1) showsthat 7) isequivalentto(7') Vx > 1 limtO. R(tx) - R(t)}/(2R(tx) og logR(tx))i = ooSuppose forsome x > 1, (7') is false. Then thereexist s > 0, yn ?? and a,0 < a < oo such that:(13) limn.-oR(y.) - R((1 - e)y.)}/(R(y.) og logR(y.))' = a.We can and do supposethat thesequence {yn} s so thin hatR(yn_)/R(yn)O 0.Set h(y)) = {R(y) - R((1 - s)yJ/R1(y)o thatfrom 13) we have

h(yn)/(logogR(yj)) aand henceh(y.)/log RI(y.) - 0.So forgiven C, 0 < C < 1, we have for sufficientlyarge n, n _ nOsay, thath(yn) < ClogRi(yn)-.Note thatRi(y) - Ri((l - s)y) < h(y). Hence, keeping nmindthat 9) entails


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656 SIDNEY I. RESNICKh(y) non-decreasing,we have:

S expI-(RI(y) - Rl((1 -)y))} dR'(y)> exp h(y)}dR'(y)> Ad Sn+in=n yn> 1=nO e-h(y+P(RI(yn+i) - RI(y,))> E noe-4 logRi(yn+i)(RI(y.+1) - RI(y))

nn=noR1(yn+1))1-C ooThis provesthefirstssertion.REMARK. This assertion an be proved finsteadof (9) one assumesthat

lims-,0 -1(s + t(s og log t)i)/Ri(s)existsfor ll real t. The significancef this ypeofassumptions explained n[6].To show that the converse in Theorem 6 is false, consider the R-functionR(x) = (log x log log x)2, x > ee. LettingM,,a) be themaxima of i.i.d. randomvariables drawnfrom he associated distribution,we can show: (i) {M.(a)} stablei.p.; (ii) {M.(a)} notstable a.s.; (iii) {Mn} table a.s.; and (iv) {XL}j stablea.s.The proof of (i) follows by noting that log (1 + s)x log log 1 + s)x -logx log logx -> oo, which implies1 - F(a)(x) is rapidlyvarying. The verifica-tion of (ii) comes from showingthe integral n (12) diverges. To prove (iv),show that 7) holds. This isdonemosteasily by verifyinghe sufficientondition(see [6] Theorem 2):

limrO.OR'(x)/(R(x) log log R(x))i = oiFinally (iv) (iii) by Theorem5.We nextconsider .s. stability f I Al3)j}. Our result s related o a theoremofGrenander 11] concerning tability .p. of successive sums of minima froman i.i.d. sequence. An a.s. versionof the Grenanderresultwas later proved by0. Frank [9]. The author is gratefulto Dr. H. Cohn forpointingout thesereferences.THEOREM7. If {M.} is a.s. stable, o is {I, =1Mj}:

lim,. E j> Mj/,>j1 R-1(log ) = 1 a.s.If in addition R-l(x) is regularly arying xponent , 0 < a < oo (equivalentlyR(x) is regularlyvarying xponenta-1 via [2], Corollary 1.2.1), then a.s.limno, E j> Mj/nR-'(logn) = 1.

PROOF. The first ssertionfollows from the following nalytical lemma: Ifan > 0, b. > 0 forsufficientlyarge n, E a. = oo and limn_,../b, = 1, thenlimn_4. j> aj/ILn=jb3= 1. For the second result note that f R-1 is regularlyvarying, henR-1(logx) is slowlyvarying ince logx is slowlyvarying [2] page21). Hence

Eij R-1(log j) sn R-1(log x) dx ~ nR-1(log n)as n -+ oo via [8] page 281 or [2] page 15.


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RECORD VALUES AND MAXIMA 657The Grenanderresultdoes not seemto come from hesemethods, pparentlydue to the impossibilityffindingonstantsbe uchthatMa - b- 0 as n -> ooa.s. (or even i.p.-see [10]) for the case of maxima drawnfrom negativeex-

ponentialdistribution. nsteadwe obtainthefollowingdisjointregult:THEOREM 7'. X1,X2,* are i.i.d. withcommon ontinuous istribution(x).DefineZn= min X1, * , X} and Q(x) = -log F(x). SupposeQ-l(x) = x-aL(x)as x -+ oowhere o > a > 0 andL is slowlyvarying.Then(i) lim,.o ZJQ-1(log n) = 1 a.s., and(ii) limn0. E l Zj/nQ-'(logn) = 1 a.s.4. Comparison f thedomains f attractionf the imitingecordvalue distribu-

tionsand the extreme alue distributions.When will both {XL } and {MJ}havelimitingdistributions? That is, for a distribution (x) with R-functionR(x),whenit is truethatF(x) C DM(G(x)), for some extremevalue distributionG(x),and R(x) C DR(H(x)) for some limiting ecord value distributionH(x).In thissectionwe will designate hethree imiting ecord value distributionsby N(x), Nla(x) and N2a(x) where N(x) is the standardnormal distribution,Nla(x) = N(-log (-log Da(x))), N2,(X)= N(-log (-log Vla(x))) where DFa(x)and V1a(x) are theextremevalue distributions ivenafter 2).In thecase that M.} has limit distributionG(x) and {XL)} has limit aw H(x)we will say thatthepair (F(x), R(x)) is dually ttracted o (G(x), H(x)) and write(F(x), R(x)) C D(G(x), H(x)) .

THEOREM 8. Theonly ossibilitiesor dual attractionre:(F(x), R(x)) C D(A(x), N(x))(F(x),R(x))C D(A(x), Nla(x))(F(x), R(x)) C D(A(x), N2a(X))

Thus necessaryonditionordual attractions thatF(x) C DM(A(x)).PROOF. We use freely hefactsobtained from heduality:

R(x) C DR(N(-log (-log G(x)))) iff 1 - e-RI(x) C DM(G(x)) .Suppose R(x) ? DR(Nia(x)). Then ([16] Theorem4.2) for all t > 1:

limx,. {R(tx) - R(x)}/Ri(x) = a logtand hencelima R(tx) - R(x) = oo.This precludes1 - F(x) beingregularly arying nd thusF(x) i DM(QDa(x))forall a > 0. Also R(x) C DR(Nia(x)) implies that F(x) < 1 for all x and henceF(x) q DM(Va(x)) since distributionsn the domain of Vla(x) must have finiteright nd.The demonstrationthat R(x) e DR(N2a(x)) implies F(x) q DM(Da(x)) and


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658 SIDNEY I. RESNICKF(x) q DM(VJa(x)) s almostthe same as the previouscase. Now assume R(x) eDR(N(x)). Then 1 - e-RIx, e DM(A(x)) whichimplies f the right nd is infinitethat -Ri'z) is rapidlyvarying:

e- Ri(t)/le-Ri(x) O 0for all t> 1as x -- oo. By Theorem2 thismeans thatlim,_. {R(tx) - R(x)}/Ri(x) = 00

forall t > 1 and thisprecludestheregularvariationof 1 - F(x) so thatF(x) 1DM($,a(x)) forany a > 0. The remaining ases are disposed of in a similarmanner.Before continuingwe make thefollowing onventions: Let F(x) be a distri-bution with R-functionR(x) and right nd x0 x0< oo). Suppose the firstndsecondderivatives fR(x) exist n someneighborhood fx0and are denotedbyr(x) and r'(x) respectively.Then

(i) F(x) is a VonMisesfunction f typeA(x) if ultimatelyr(x) > 0 and if(1/r(x))' + 0 or equivalently fr'(x)/r2(x)O 0 as x -> x0.(ii) F(x) is a VonMisesfunctionf type ,(x) ifxO= ooand there xists > 0such thatxr(x)-- a as x -* oo.(iii) F(x) is a VonMisesfunctionf typeV1Ia(x)fx0< ooand there xists > 0such that xO x)r(x) -* a as x TxO.It is known see [9] or [2] pages 109-112) that (i), (ii), (iii) are respectivelysufficientorF(x) e DM(A(x)), F(x) ? DM(Ij(x)), F(x) ? DM(QJa(x)).Returning o the contentof Theorem 8 let us show that the intersection fthe domainsof attractionofA(x) and the limiting ecordvalue distributionssindeednonempty. To do this,considerthe class of R-functions iven by

Ra(X) -( log x)for x > e and a > 0, , > 0. Call the corresponding istribution j(x). Thenthefollowing onditions uicklyemerge:

(i) For a > 1, Fa(x) is a Von Mises function f typeA(x).(ii) For a = 1, F1(x) is a Von Mises functionof type D,32(x). This followsimmediately rom he factthatxr,(x) = ,/2.(iii) For a < 1, Fa(x) i DM($Dr(x))forany r > 0 and Fa(x) iZDM(A(x)).The first ssertionfollowsquicklyby showing1 - Fa(x) is rapidlyvarying ndhence not regularlyvarying. The second assertion an be provenby usingthecriterion or attraction o A(x) given n [2] page 76 or [13].From (i), (ii) and (iii) and theDuality Theorem [16] Theorem 4.1) we havethe following:

(i) For a > 2, Ra(X) ? DR(N(x)), Fa(X) ? DM(A(x)).(ii) For a = 2, R2(x)? DR(N1,(x)), F2(x) ? DM(A(x)).


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RECORD VALUES AND MAXIMA 659(iii) For 1 < a < 2, Rar(x) DR(Na,(x)) for nyr, Far(x) DM(A(x)),For 1 < a < 2, Ra(x) q DR(N(x)).(iv) For a = 1, R_(x)VDR(N,,(x)) forany r, F1(x)CM(? ,(x

For a = 1,R_(x)VDR(N(x)).Thus we see thatDM(A(x)) n DR(N(x)) # 0, and DM(A(x)) n DR(N,,(x)) + 0for any ,3> 0 and by symmetry onsiderations t is clear that DM(A(x)) nDR(N2,(x)) / 0. Further t s possiblethat{Mn}has a limiting istribution utnot {XL IOur mostpreciseresults bout theoverlapof thesedomains of attraction recontained n thefollowing:THEOREM . Let F(x) be a distributionithR-function(x) and associated istri-

bution (a)(x) = 1 - e-A(x)(i) If R(x) C DR(N(x)) and F(a)(x) is a Von Misesfunction f typeA(x), thenF(x) is a VonMisesfunction f typeA(x) andF(x) e DM(A(x)).(ii) If R(x) ? DR(Nia(x)) and F(a)(x) s a Von Misesfunction f type D1/2(x),then (x) is a VonMisesfunctionftypeA(x) andF(x) ? DM(A(x)).(iii) If R(x) ? DR(N2a(x)) and F(a)(x) is a VonMisesfunction f typeV11a2(x),then (x) is a VonMisesfunction ftypeA(x) andF(x) ? DM(A(x)).PROOF. Because of theDualityTheorem [16] Theorem .1) it sufficesoshow:If F(x) is a Von Mises function f type A(x), (Da(x) or 1ffa(x)hen1- e-R2z) isVon Mises of typeA(x). To prove thiswhenF(x) is a Von Mises function ftypeA(x) note (R2(x))' = 2R(x)r(x) and

(1= r (x)1 0 as x- oo.(R2(x)) 2R(x)r2(x) 2R(x)In the case that F(x) is a (I>a(x)typeVon Mises functionwe have r(x) ax-'.Hence r'(x) -ax2 as x oo ([2] page 23). But:

r'(x)+ 1(R2(x))' Mr2(x) 2R(x)The second termgoes to zero and thefirst erm s asymptotic o

-ax -2 1 02R(x)a2X-2 2aR(x)as x -> oo. The third ase is handledin a similarmanner nd this ompletes heproof.

We have alreadyseen that t is possiblefor{M.} to have a limitingdistribu-tion but not {XLJ}. The conversesituation an holdas well: It can happenthatR(x) e DR(H(x)), forsome limitingrecord value distributionH(x), but F(x) isnot attracted to any extreme value distribution. Thus Theorem 9 cannot beextended to all continuousdistributions.This is surprisingnviewofthe recentde Haan-Balkema result 5] which states that if F(x) ? DM(A(x)) thenF(x) is


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660 SIDNEY I. RESNICKclose to a A(x)-typeVon Mises function ,(x) in thesense that 1 - F(x))/(l -Fj(x)) -> 1 as x -+ x0 x0 s theright nd ofF(x)).The following xample by Laurens de Haan shows the possibility hatR(x) CDR(N((x)) but F(x) i DM(A(x)) (and hence by Theorem8 F(x) is not attractedto any extremevalue distribution). By the DualityTheorem [16] Theorem4.1)it suffices o exhibit distribution (x) such thatF(a)(x) GDM(A(x)) but F(x) 1DM(A(x)). DefineF(a)(x) by 1 - F'a)(x) = e-R1(x) whereRi(x) = xi + 2x-* sinx,x _1. ThenRI(x)-xi 0 as> so 1-Fa(X) - e- ?s x-oo. Sincethe distribution - e-zi is a Von Mises function f type A(x) we have by tailequivalence [15] that F'a)(x) e DM(A(x)).SinceR(x) = x + sin x + (sin2 )/4x nd R(x) - (x + sinx) -> 0 as x + oo wehave that F(x) is tail equivalent to F1(x) = 1 - exp (x + sinx)} and F(x) eDM(A(x)) iffF1(x) e DM(A(x)). Hence it suffices o show F1(x)VDM(A(x)).Proceedingby contradictionwe have that ifF1(x)e DM(A(x)) then thereexistsan auxiliaryfunction (t) ? 0 such that

limt.., R1(t xf(t)) - R1(t)= xfor all x whereR1(t)= -log (1 - Fl(t)). The above relationmusthold alongthe sequence t, = 2n7rwhere by the periodicityof sinx we have thatRl(tn+xf(tQ)) R1(t.) = xf(tn) sinxf(tn). Take a further ubsequence t,' such thatf(tt,')+ c e [0, oo]. A contradictions obtainedby showingtheincompatibilityof the following elations:

Rl(tn'+ xf(tn)) - Rl(tt') >xc + sin xcRl(tn' + xf(tn')) - Rl(tn') x

Ifc = 0 or oo the contradictions clear interpretingimn . xf(tn sinxf(tn) oofor the case c = oo). If 0 < c < oo we have for all x that x = xc + sin xc orsinxc = x(1 - c) which is not true. This completesthecounterexample.To close thisdiscussion of limit aws we present ome results bout theshapeof R(x) when R(x) is attracted o some limiting ecord value distribution.THEOREM 10. SupposeR(x) ? DR(H(x)) whereH(x) is one of the three imitingrecordvalue distributions. henthere xistsa type-A(x)VonMisesfunction *(x)with orresponding-function*(x) suchthat

F*(x) ? DM(A(x)) and R(x) , R*(x)as x -+ xO xO s theright ndofthedistribution).

PROOF. If R(x) e DR(Ni,(x)) thenby dualitywe have:R(x) - R*(x) = logx)(see Remarksfollowing heorem4.2, [16]). It isquickly checkedthatthisR*(x)is a Von Mises function f typeA(x). Similar techniquesprove theresultwhenR(x) e DR(N2ar(x))o now supposeR(x) ? DR(N(x)) and theright ndofF(x) isx,.


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RECORD VALUES AND MAXIMA 661Duality shows that F(a)(x) e DM(A(x)) and hence from 5] Fla)(x) is tail equiva-lent to a A(x)-typeVon Mises function; .e., thereexist c(x), r(x) such thatc(x) -- 1 and r'(x)/r2(x) 0 as x > oo and

1- Fla(x) - c(x) exp{I- sx r(t) dt}SetR*(x) = ( X r(t) t)2. Since

Ri(x) - -log (1 - F aI(x)) -log c(x) + $X.r(t)dt x$ r(t)dtwe musthave R(x) ~ (S Xr(t)dt)2 R*(x) .But 1 - exp - S 0.r(t)dt) is a Von Mises functionof type A(x) and hence byTheorem 9 so is R*(x). The proof s complete.Acknowledgments. uch of thiswork was done while theauthor was a sum-merguestof the Mathematics nstitute f theUniversity f Amsterdam nd theAmsterdamMathematicsCenter. The authorthanks hose nstitutions or theirhospitality nd support.The initial draft f this paper was completedwhiletheauthorwasworking ttheTechnion-Israel nstitute fTechnology.Wim Vervaatmade severalhelpful uggestionswhichwere ncorporatedn thefinalversion.

REFERENCES[1] BARNDORFF-NIELSEN, . (1963). On the imitbehavior fextreme rder tatistics.Ann.Math. Statist.34 992-1002.[2] DE HAAN, L. (1970). On Regular Variation nd Its Application o the Weak Convergence fSample Extremes. Mathematical Center Tracts 32, Amsterdam.[3] DE HAAN, L. and HORDIJK,A. (1972). The rate of growthofsample maxima. Ann.Math.Statist.43 1185-1196.[4] DE HAAN,L. (1972). Equivalence classes ofregularlyvaryingfunctions. Technical ReportNo. 41, Department of Statistics, StanfordUniv.[5] DE HAAN,L. and BALKEMA,A. (1972). On R. Von Mises' condition for the domain of at-traction of exp{-e-x}. Ann. Math. Statist.43 1352-1354.[6] DE HAAN,L. and RESNICK, S. I. (1972). Almost sure limitpoints of recordvalues. Tech-nical Report No. 39, Department of Statistics, Stanford Univ. To appear J Appl.Probability.[7] FELLER, W. (1946). The law of the iterated logarithm for identicallydistributedrandomvariables. Ann.of Math. 47 631-638.[8] FELLER, W. (1966). An Introduction o Probability Theory nd Its Applications2. Wiley,New York.

[9] FRANK, 0. (1966). Generalization f an inequality fHajek andRenyi. Skand. Aktuari-etidskr. aft1-2,85-89.[10] GNEDENKO, B. V. (1943). Sur la distributionimitedu termemaximum 'une serie lea-toire. Ann. of Math. 44 423-453.[11] GRENANDER, U. (1965). A limit theoremfor sums of minima ofstochastic variables. Ann.Math. Statist.36 1041-1042.[12] HARTMAN, P. and WINTNOR, A. (1941) On the law of the iterated logarithm. Amer. J.Math.63 169-176.


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662 SIDNEY I. RESNICK[13] MARCUS,M. and PINSKY, M. (1969). On thedomain f ttractionof exp{-e-x}. J. Math.Anal. Appl. 28 440-449.[14] RMNYI,A. (1962). Theoriedes elements aillants 'une suited'observations. olloquiumon CombinatorialMethods nProbability heory,August1-10,Mathematisknsti-

tut,AarhusUniversitet,enmark.[151RESNICK, S. I. (1971). Tail equivalence nd ts pplications.J.Appl. robability136-156.[16] RESNICK, S. I. (1973). Limit aws forrecordvalues. J. Stoch. Processes and TheirAppl.1 67-82.[17] RESNICK,S. I. and TOMKINS,R. J. 1972). Almost ure stabilityf maxima. To appearJ. Appl. Probability.[18] TATA,M. N. (1969). On outstanding alues na sequence f random ariables.Z. Wahr-scheinlichkeitstheoriendVerw. Gebiete12 9-20.[19] VON MISES,R. (1936). La distributione la plus grandede n valeurs. Selected Papers 2.American Mathematical Society, Providence, 271-294.DEPARTMENT OF STATISTICSSTANFORD UNIVERSITYSTANFORD, CALIFORNIA 94305

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