MAXIMA OF POISSON-LIKE VARIABLES AND RELATED TRIANGULAR ARRAYS 1

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1 The Aals of Applied Probability 1997, Vol. 7, No. 4, MAXIMA OF POISSON-LIKE VARIABLES AND RELATED TRIANGULAR ARRAYS 1 By Clive W. Aderso, Stuart G. Coles ad Jürg Hüsler Uiversity of Sheffield, Uiversity of Lacaster ad Uiversity of Bere It is kow that maxima of idepedet Poisso variables caot be ormalized to coverge to a odegeerate limit distributio. O the other had, the Normal distributio approximates the Poisso distributio for large values of the Poisso mea, ad maxima of radom samples of Normal variables may be liearly scaled to coverge to a classical extreme value distributio. We here explore the boudary betwee these two kids of behavior. Motivatio comes from the wish to costruct models for the statistical aalysis of extremes of backgroud gamma radiatio over the Uited Kigdom. The methods exted to row-wise maxima of certai triagular arrays, for which limitig distributios are also derived. 1. Itroductio. Oe result of the public cocer aroused by the Cherobyl accidet i 1986 was the settig up i several Wester Europea coutries of a etwork of idepedet moitorig statios for backgroud gamma radiatio [the Argus Project: see E Garde 1993]. I the Uited Kigdom more tha 15 moitorig statios have ow bee i operatio for several years. Each cotiuously records the arrival of γ-rays ad oce a day dowloads aggregated 10-miute couts to a cetral data bak. The large volume of data thus accumulated offers a uprecedeted opportuity to explore the temporal ad spatial patters of variatio of backgroud radiatio. Of particular iterest i ay aalysis of the data are the uusually high values of radiatio, sice the occurrece of exceptioally high levels may be idicative of some further accidetal uclear emissio. This motivates the search for statistical models o which to base a aalysis of the spatial ad temporal characteristics of extremes of backgroud gamma radiatio. The physical laws which gover the behavior of radiatio emissio suggest that couts over fixed periods should follow a Poisso law. Fluctuatios i both meteorology ad atmospheric coditios as well as imperfectios i recordig devices cause modificatios i this basic law, resultig i ostatioarity ad perturbatios i the margial Poisso behavior. The aim of this paper is to develop a framework for modellig the extremal behavior of a sequece of Poisso variables, which is robust to misspecificatio of the margial Poisso distributio. This falls i the ambit of classical extreme value theory: give a sequece of idepedet variables X 1 X with commo distributio fuc- Received December 1994; revised April Part of research used equipmet fuded by SERC uder Complex Stochastic Systems Iitiative. AMS 1991 subject classificatios. Primary 60G70; secodary 60F10. Key words ad phrases. extreme values, Poisso distributio, large deviatios, triagular arrays, regular variatio, subexpoetial distributios, modellig of extremes, radiatio couts. 953

2 954 C. W. ANDERSON, S. G. COLES AND J. HÜSLER tio F, a sequece of ormalizig fuctios u x is sought such that 1 P max X i u x G x 1 i where G is a odegeerate distributio fuctio. I applicatios, F is geerally ukow, but the class of possible limits G, usually referred to as the extreme value family, is sufficietly arrow to permit modellig of max 1 i X i directly as G. I the case of Poisso variables with mea λ, this argumet fails. Aderso 1970, 1980 studied the case where the X i are Poisso variables ad foud that there is a sequece of itegers I for which lim P max 1 i X i = I or I + 1 =1, so that o ormalizig fuctios u x ca be foud which lead to odegeerate limits i 1. Figure 1 illustrates this behavior for λ = 2. As icreases, the distributio of max 1 i X i cocetrates icreasigly o a pair of cosecutive itegers. The asymptotic properties of the sequece of itegers I have bee characterized by Kimber This argumet gives o justificatio therefore for modellig Poisso maxima through extreme value distributios. The theme of this paper is the presetatio of a argumet which evertheless justifies the use of extreme value distributios for modellig Poisso maxima whe λ is sufficietly large. Our reasoig is as follows: for large λ the Poisso distributio ca be approximated by a Normal distributio. If Fig. 1. Distributio of the maximum of = 10 k idepedet Poisso radom variables with mea 2.

3 MAXIMA OF POISSON-LIKE VARIABLES 955 the X i i 1 are Normal, it is well kow that u ca be foud so that the Gumbel limit G x =exp e x is obtaied. Cosequetly, usig the Normal approximatio first ad the applyig 1, we obtai a Gumbel approximatio to the distributio of Poisso maxima. This limitig behavior is formalized i Sectios 2 ad 3 ad supported by umerical calculatios i Sectio 5. The argumet higes critically o the relative rates of covergece of the Poisso Normal limit ad the extreme value limit i 1. The sharpess of the coditio for a Gumbel limit is show to deped o the umber, r, of terms used i a series expasio for the tail behavior of the Poisso distributio. I Sectio 6 it is show too that if the Normal covergece is too slow, the the stadard degeerate behavior of Poisso maxima persists. Our argumets for a Gumbel limit do ot i fact deped critically o the radom variables beig Poisso. I Sectio 4 we show how they exted to row-wise maxima of certai triagular arrays of variables, each covergig i distributio to ormality. We also preset some further results about maxima of triagular arrays, which show that i o-poisso heavy-tailed cases, both the Gumbel ad Fréchet extreme value distributios ad a related oextreme value distributio may also arise as limits. 2. The mai limit result. For each positive iteger, let R i i = 1 deote idepedet Poisso radom variables with mea λ growig with. We study max 1 i R i as.asλ grows, the Poisso distributio of each R i approaches ormality, ad so we might expect that for rapidly icreasig λ, ormality would set i quickly eough for the distributio of max 1 i R i to resemble that of the maximum of idepedet Normal variables. The results to follow show that this is ideed the case, ad they give appropriate growth rates for λ which guaratee it. The questio we address is the followig: whe is it possible to fid fuctios u x ad a odegeerate distributio fuctio G x such that 2 P max R i u x 1 i G x as, ad what forms do u ad G the take? Sice the R i are idepedet, 2 is equivalet to 3 lim P R 1 >u x = log G x ad this is the expressio we maily work with. A estimate of the probability i 3 may be obtaied from the large deviatios results of cetral limit theory. The mai result used here, Cramér s theorem [see, for example, Petrov 1975, page 218], applies to idepedet idetically distributed radom variables X i whose momet geeratig fuctio exists i a eighborhood of the origi. If E X i =0 Var X i = σ 2 ad S = 1 X i, the for x varyig with i such a way that x ad x = o 1/2, P S /σ 1/2 >x x 4 = exp x 2 C 1 x 1/2 [ ] x 1 + O 1/2

4 956 C. W. ANDERSON, S. G. COLES AND J. HÜSLER where C is a power series C z =c 1 z + c 2 z 2 + whose coefficiets are determied by the momets of the X i, c j beig a fuctio of momets of order j + 2 ad lower. We apply this result iitially to cetered uit Poisso variables X i, replacig by λ, so that S i 4 follows the same cetered Poisso distributio as R i λ. Thus 5 R 1 λ P >x 1 x exp x 2 C x /λ 1/2 whe x = o. The first coefficiet of C i this case, for example, is c 1 = µ 3 /6σ 3 = 1/6 µ 3 beig the third momet of X i. This argumet appears to require that the sequece λ should be iteger valued, but i fact a check o the proof of Cramér s theorem shows that it works also for cotiuously varyig λ. By takig the first r 0 terms of the C series, we see from 5 that 6 { R 1 λ P >x 1 x exp c 1 x 3 x r+2 + +c r λ r/2 for x r+3 = o λ r+1 /2. Thus the covergece 2 will certaily occur if it is possible to fid u x satisfyig 7 ad 8 u x λ { 1 x exp c 1 x 3 = x = o λ r+1 / 2 r+3 x r+2 } + +c r log G x λ r/2 We prove the existece of x ad G satisfyig 7 ad 8 i two steps. First we show that if there is a sequece β for which { β 3 β r+2 } 9 1 β exp c 1 + +c λ 1/2 r 1 λ r/2 the there is a positive sequece α for which 8 holds with x = α x + β for each fixed x. The we show that a β satisfyig 9 ca always be foud. Proofs of these facts are cotaied i Lemmas 1 ad 2 i Sectio 3. It is show moreover i Lemmas 1 ad 2 that, whatever the value of r, for x of this form, the limit i 8 is log G x =e x so that G is a Gumbel distributio; ad for each fixed x, x β 2 log 1/2 }

5 MAXIMA OF POISSON-LIKE VARIABLES 957 It follows that 7 is satisfied provided log = o λ r+1 / r+3 ad so the covergece i 2 occurs whe λ grows faster tha log 1+2 r+1 1. Our mai techical fidigs may therefore be summarized i the followig. Propositio 1. Let R i deote idepedet Poisso radom variables with mea λ, ad suppose that λ grows with i such a way that for some iteger r 0, The there is a liear ormalizatio log = o λ r+1 / r+3 such that u x =λ + λ 1/2 β r + α x lim P max R i u x = exp e x 1 i The costats α ad β r are specified more fully i the followig sectio. For the special case whe r = 0, these facts may be deduced from the large deviatios result 6 ad kow extreme value properties of the Normal distributio. We briefly outlie the details for later use. Whe r = 0, relatio 6 becomes R 1 λ P >x 1 x for x = o λ 1/6, ad it is well kow [see, e.g., Leadbetter, Lidgre ad Rootzé 1983, page 14] that for ad 10 Thus, provided β 0 lim 1 α x + β 0 = e x α = 2 log 1/2 = 2 log 1/2 log log + log 4π 2 2 log 1/2 2 log 1/2 α x + β 0 = o λ 1/6 covergece to G x =exp e x i 2 will hold. The coditio o λ for this case is evidetly that it should grow faster tha log 3.

6 958 C. W. ANDERSON, S. G. COLES AND J. HÜSLER 3. Proofs. Lemma 1. Suppose that for fixed r there exists a sequece β = o for which 9 is true. The there exists a sequece α > 0 for which 11 { 1 x exp c 1 x 3 for each fixed x, with x = α x + β x r+2 + +c r λ r/2 } e x Proof. O takig logs i 11 ad usig the fact that, as x, 1 x exp x 2 /2 / x 2π we see that it will be eough to prove x 2 12 lim 2 + log x log 2π x2 C r x / log = x where C r deotes the trucated series C r z = r c j z j j=1 Write h x =x 2 /2 + log x + 1/2 log 2π x 2 C r x/λ 1/2. By assumptio lim h β log = 0 so 12 will be proved if we show 13 lim h α x + β h β =x To prove 13 let µ x =h x. The µ x =x + x 1 xp x where p x = r 1 j + 2 c j x/ j, ad so, sice β ad β = o, It follows that β + µ β β x µ β β ad that x 14 µ β + µ µ β β each uiformly over compact sets of x. From the mea value theorem, for fixed x, h β + x µ β h β =x µ β + ξ /µ β µ β

7 MAXIMA OF POISSON-LIKE VARIABLES 959 for some ξ lyig betwee 0 ad x. By the uiformity i 14 the right-had side coverges to x, ad 11 is proved, with α = 1/µ β. Lemma 2. If log = o λ, the for each r 0 the equatio 15 h x = x2 2 + log x log 2π x2 r j=1 x j c j = log has a solutio β r with the property 16 β r 2 log 1/2 Moreover β r is the oly solutio of 15 i the regio x = o. Proof. We first prove existece of a solutio β r of h x =log i the regio x = o, ad the show that β r must satisfy 16 ad is uique. To start, ote that if x = o ad x the h x x 2 /2. Let ρ = log /λ. The λ ρ 1/2 = o λ ad log = o λ ρ 1/2, so that h ρ1/4 λ ρ 1/2 /2 > log for large eough. O the other had, for ay fixed x 0,ifis large eough, h x 0 < log But h x is cotiuous, so for each large eough, the equatio h x =log has a solutio i the iterval x 0 ρ 1/4. Moreover, ay such solutio β r must satisfy log = h β r β r 2 /2 ad so 16 must hold. Uiqueess is established by showig that, for large eough, h x is strictly icreasig i ay iterval of the form x 0 λ 1/2 ε, where x 0 > 0is arbitrary ad ε > 0 coverges to 0. This follows, for example, from the easily verifiable fact that lim uiformly i such a iterval. h x x 1 1 x 2 = 0 Remark 1. We ote from 16 that the scalig costats α defied i Lemma 1 satisfy 17 α = 1/µ β r 2log 1/2

8 960 C. W. ANDERSON, S. G. COLES AND J. HÜSLER β 0 Remark 2. For explicit expressios for β r deote the locatio costat 11 for the r = 0 case, ad set β r The, expadig h β 0 β 0 2 r 1 0 β j c j + δ β δ log about β 0,wefid + 1 β 0 β 0 we ca argue as follows. Let = β 0 +δ. 0 r β j + 2 c j whece β 0 r1 c j β 0 /λ 1/2 j δ = 1 r 1 j + 2 c j β 0 / j + o 1/β r = β 0 β j 0 r β c j 1 + j + 2 c j 1 1 j 1 + o β 0 j + o δ =0 Oly terms oegligible i compariso to log 1/2 eed be retaied i this expressio, sice terms of order o α will ot affect the limitig distributio [Feller 1971, page 253]. The above gives a first correctio to β 0. I priciple, further correctio terms may be foud by expadig aroud the ew approximate β r ad retaiig oly terms oegligible i compariso to log 1/2. For r = 1wefid ad for r = 2, β 2 β 1 = 2 log 1/2 = 2 log 1/2 log log + log 4π 2 2 log 1/2 log log + log 4π 2 2 log 1/2 + 2 log 1/2 c 1 2 log 1/2 For the uit Poisso, c 1 = 1/6 ad c 2 = 1/8. 2 log + c 1 λ 1/2 + c 2 + 3c1 2 log 2 λ 4. Maxima of triagular arrays. The result i Propositio 1 may usefully be viewed i the wider cotext of the geeral theory of maxima of triagular arrays. A cetral problem i this theory is as follows. Suppose that we are give a triagular array of radom variables S i i = 1 = 1 2, idepedet ad idetically distributed i each row, ad with commo distributio fuctio F i the th row. If the row distributios F coverge weakly to some odegeerate limit H as, what are the possible odegeerate limit distributios, G, say, for max i S i b /a for suitable costats a > 0 ad b, ad whe does covergece to a specific G occur?

9 MAXIMA OF POISSON-LIKE VARIABLES 961 For this geeral problem, it is clear that the class of limit distributios G cotais the extreme value distributios. Also, a simple sufficiet coditio for covergece of max i S i b /a to a extreme value limit G is evidetly that H should belog to the max domai of attractio of G ad that covergece of F to H should be fast eough i the upper tail. A specific coditio for the latter by a argumet similar to that i the r = 0 case at the ed of Sectio 2 is that, for each τ>0, 19 1 F s 1 H s for sufficietly large s y τ, where y τ satisfies 1 H y τ τ/ as. Propositio 1 goes beyod this simple result i the special case of scaled Poisso variables S i by showig that a weaker coditio tha 19 ca hold for them ad still be sufficiet for a Gumbel limit G. Moreover the argumet leadig to Propositio 1 uses the Poisso ature of the variables oly to guaratee the applicability of Cramér s theorem 4, ad so the coclusio of Propositio 1 ca be expected to hold i other cases whe F is a covolutio. What is required is that each variable S i should be represetable as a sum, suitably scaled, of idepedet ad idetically distributed radom variables whose momet geeratig fuctio exists i a eighborhood of the origi. Specifically, let U j j 1deote i.i.d. radom variables whose momet geeratig fuctio exists i a ope eighborhood of the origi, ad suppose that for some sequece of itegers k, d / S i = U j c k d k j k where c k = kµ ad d k = σk 1/2 with µ = E U 1 ad σ 2 = Var U 1. The, by the same argumets as led to Propositio 1, we have Propositio 2. Propositio 2. For each positive iteger, let S i i = 1 deote idepedet radom variables, each of which is a sum, scaled to zero mea ad uit variace, of k idepedet ad idetically distributed radom summads whose momet geeratig fuctio exists i a ope iterval cotaiig the origi. If log = o k r+1 / r+3 for some iteger r 0, the lim P max S i α x + β r = exp e x 1 i where α ad β r are the ormalizig costats defied i the previous sectio. Though Propositio 2 suffices for the immediate eeds of our modellig problem see Sectio 6 below, it is iterestig to explore the limitig behavior of maxima of triagular arrays more geerally. I Propositio 2 the existece of a momet geeratig fuctio is somewhat restrictive, excludig heavytailed distributios. To ivestigate a heavy-tailed case i which the S i still coverge to ormality, suppose that E U 1 2+δ < for some δ>0 ad that the

10 962 C. W. ANDERSON, S. G. COLES AND J. HÜSLER distributio fuctio, K, say,ofthe U j µ /σ has regularly varyig tail, K α for some α>2. The, for each i, S i properly ormalized coverges still to a ormal radom variable, but the momet geeratig fuctio coditio is ot satisfied. To study this case we use a large deviatios result of A. V. Nagaev 1969a [see also S. Nagaev 1979], which shows uder the coditios above that 20 k P S 1 >x =P U i >k µ + k 1/2 xσ 1 = 1 x 1 + o 1 + k K k 1/2 x 1 + o 1 for k ad x 1. The followig heuristic argumet based o 20 idicates the kid of limit distributios ow to be expected for max i S i. Multiplyig 20 by ad takig expoetials suggests that for large x 21 P max S i x i x K k k 1/2 x so that a limitig distributio for max S i might be expected to coicide with a limit distributio for the maximum of two idepedet radom variables, oe of which is the maximum of idepedet stadard ormal variables ad the other the maximum of k idepedet copies of U j µ /σk 1/2. Let the sequece b k be such that k K b k 1ask. Loosely speakig, b k is a measure of the locatio of the distributio of max i k U i µ /σ. It follows that b k /k 1/2 is a measure of the magitude of max j k U j µ /σk 1/2, the secod radom variable i our iformal iterpretatio above. Similarly 2 log 1/2 is approximately the order of magitude of the maximum of i.i.d. stadard ormal radom variables, the first term i the iformal iterpretatio. Suppose 22 2 log 1/2 b k /k 1/2 x 0 as. A value of x 0 = suggests that i 21 the ormal maximum will domiate, ad so max S i will coverge to a Gumbel distributio: a value x 0 = 0 o the other had suggests that the term based o the maximum U j will domiate, ad so a Fréchet limit distributio will result. The followig propositio makes these rough argumets precise, ad clarifies the behavior whe 0 <x 0 <. I the latter case a oextreme value limit is foud. Propositio 3. For each positive iteger, let S i i = 1 deote idepedet radom variables, each of which is a sum, scaled to zero mea ad uit variace, of k idepedet ad idetically distributed radom summads U j with distributio fuctio K. Suppose that E U j 2+δ < for some δ>0, that K α for some α>2, ad that 22 holds for some x 0.

11 i If x 0 = the lim P ii If 0 x 0 <, the Proof. MAXIMA OF POISSON-LIKE VARIABLES 963 max i lim P max S i b k i x/k 1/2 = S i α x + β 0 = exp e x { exp x α for x x 0 0 for x<x 0 i Suppose x 0 =. The for ay real x ad ay B>0, α x + β 0 evetually. From 20, P S 1 >α x + β 0 β 0 = 2 log 1/2 >Bb k /k 1/2 = 1 α x + β o 1 + k K k 1/2 α x + β o 1 The first term o the right here coverges to e x while the secod is bouded above by k K Bb k B α which ca be made arbitrarily small by choice of B. Thus lim P S 1 >α x + β 0 = e x which proves the assertio. ii Suppose x 0 <. Ifx>x 0, the xb k /k 1/2 B 2 log 1/2 evetually for some B>1, ad so 1 xb k /k 1/2 is evetually bouded above by 1 B 2 log 1/2, which teds to 0 as. It follows that the secod term i 20 domiates, ad that P S 1 >xb k /k 1/2 k K b k x x α If x<x 0, the xb k /k 1/2 θ 2 log 1/2 evetually for some θ<1, ad so, by 20 agai, evetually P S 1 >xb k /k 1/2 1 θ 2 log 1/2 These two limits prove the assertio. For a lighter-tailed case itermediate betwee those of Propositios 2 ad 3, suppose that the U j are oegative ad that U j µ /σ has a absolutely cotiuous distributio with probability desity fuctio satisfyig 23 K x exp x 1 ε as x, for some ε 0 1. The K x x ε exp x 1 ε / 1 ε

12 964 C. W. ANDERSON, S. G. COLES AND J. HÜSLER so that the tail decays faster tha a regularly varyig fuctio, but the momet geeratig fuctio coditio of Propositio 2 is ot satisfied. The fuctio K is a member of the subexpoetial family of distributios. A defiig property of this family [Embrechts ad Goldie 1980] shows that for each fixed k, k 24 P U j kµ>k 1/2 xσ P max U j µ>xσ k K x 1 j k j=1 as x. Nagaev 1969b, Theorem 3, proves that for distributios 23 the relatio 24 cotiues to hold whe k icreases as x, provided that k 1 ε /2ε = o x. From these facts we get Propositio 4. Propositio 4. For each positive iteger let S i i= 1, deote idepedet radom variables, each of which is a sum, scaled to zero mea ad uit variace, of k idepedet ad idetically distributed oegative summads U j with fiite mea µ ad variace σ 2. Suppose that the distributio fuctio K of U j µ /σ is absolutely cotiuous ad has desity satisfyig 23. Suppose also that k = o log 2ε/ 1 ε. The lim P max S i u x = exp e x i where u x = log k 1/ 1 ε [ + log k ε/ 1 ε ε 1 ε log 2 2 k 1 ] x log 1 ε + 1 ε 1 ε Proof. Uder the coditio o the rate of growth of k, u x log 1/ 1 ε so that k 1 ε /2ε /u x = o 1, ad therefore, by Theorem 3 of Nagaev 1969b, 24 holds with x replaced by u x. The coclusio of Propositio 4 follows directly from this by a short calculatio. The Gumbel limitig distributio i Propositio 4 arises because, for k growig slowly eough, the tail of S i resembles that of its summads, the U j. Whe k grows much faster [i fact, so that log 1+ε / 1 ε = O k ], the tail of S i ultimately resembles that of the Normal distributio [Nagaev 1969b, Theorem 1], ad accordigly we would agai expect a Gumbel limit for max S i, though with differet ormalizig costats. This fact may be established rigorously by argumets similar to those i Propositio 2 above. For itermediate rates of growth of k there are subtle combiatios of the two domiat forms of tail behavior of the S i. Nagaev 1969b gives a comprehesive discussio, from which further limitig results for max i S i may be deduced. I all cases, though for the differet reasos outlied above, it is foud that the Gumbel limitig distributio exp e x persists.

13 MAXIMA OF POISSON-LIKE VARIABLES 965 Suppose ow that we drop the assumptio of a ormal limitig distributio for S i as, ad istead assume that S i = j k U j c k /d k coverges, for suitable ormalizig costats, to a stable distributio G αβ as, where 0 <α<2 ad the ocetrality parameter β lies i 1 1. Uder certai coditios o the pseudomomets of order r of the U j [see Christoph ad Wolf 1992, Sectio 5.2], we have the followig large deviatios result, which we assume to hold. For α<2 ad α<r<1 + α, / 25 P U j c k d k >x = 1 G αβ x +O k r α /α x r j k as x. The stable distributio G αβ has the well-kow tail behavior: 26 1 G αβ x cx α as x, for some costat c. A direct calculatio based o 25 ad 26 gives Propositio 5. Propositio 5. sequece k, for x>0. Uder the assumptio 25 with α<2, we have for ay lim P max S i c 1/α x = exp x α i 5. Numerical results for Poisso maxima. Figure 2 compares the distributio fuctio of a scaled versio of the Poisso maximum max 1 i R i with its limitig Gumbel distributio, for = , 1,000 ad 10,000. The Poisso meas are take to be λ = log 7/2, givig a rate of growth i the r = 0 regio of Propositio 1. Accordigly, the ormalizig costats α ad are used. The compariso is made for clarity o a double log scale, so that what is actually plotted is log log P max R i u x vs. x 1 i β 0 where u x =λ + λ 1/2 + α x. Accordig to Propositio 1, the plotted step fuctio should coverge as to the lie y = x. What is evidet from the figure is that the covergece is slow, as might have bee expected from the kow slowess of covergece of ormal maxima to a Gumbel limit. However, over the cetral rage 1 53 x 4 6, which cotais 98% of the limit distributio, the agreemet is remarkably good, eve for = 10. Figure 3 illustrates a case whe λ, here take to be log 5/2, grows more slowly with. This correspods to a r = 1 regio i Propositio 1. The step β 0 fuctio plotted i this case is based o the ormalizig costats α ad β 1. The behavior is similar to that i Figure 2, though λ reaches oly about a teth of the size.

14 966 C. W. ANDERSON, S. G. COLES AND J. HÜSLER Fig. 2. Distributio fuctio of the ormalized maximum of idepedet Poisso radom variables with mea λ = log 7/2. Double log vertical scale. Normalizatio correspodig to r = 0 case of Propositio 1. Figure 4 illustrates the eed for modified ormalizig costats i the r = 1 case. For λ = log 5/2 as i Figure 3 ad for = it compares the distributio of max 1 i R i ormalized by α β 1, with the same distributio ormalized by α β 0. Though at this value of either ormalizatio gives a perfect correspodece, the r = 1 ormalizatio does appear preferable, as would be expected from Propositio 1. Figure 5 shows the quality of covergece i relatio to the growth of λ for the r = 1 case with λ = 2 log 5/2. The plotted poits are log log P max 1 i R i u x for x = , the first, fiftieth ad ity-ith percetiles of the Gumbel distributio idicated by dashed lies o the plot. The same ormalizig sequece u x = λ + + α x is used as i Figures 3 ad 4. Oly probabilities for values of λ at itervals of 20 are plotted, but the results show both the slowess of covergece ad its oscillatory character, a cosequece of discreteess. Agai it is clear that the agreemet is closer i cetral parts of the distributio tha i the tails. This suggests that a peultimate approximatio by a o-gumbel extreme value distributio with shape parameter covergig to 0 as is likely to improve the approximatio, as it does β 1

15 MAXIMA OF POISSON-LIKE VARIABLES 967 Fig. 3. Distributio fuctio of the ormalized maximum of idepedet Poisso radom variables with mea λ = log 5/2. Double log vertical scale. Normalizatio correspodig to r = 1 case of Propositio 1. for ormal ad other maxima [Fisher ad Tippett 1928, Gomes 1994]. This ca i fact be show to be the case by argumets similar to those i Sectio The Poisso case whe o log. Whe λ grows more slowly tha log, we ow show that o limitig distributio is possible for Poisso maxima. Thus the growth coditio i Propositio 1 is close to beig ecessary as well as sufficiet for a Gumbel limit. To see this we eed some further otatio. Let R i deote idepedet Poisso variables as i Sectio 2 ad let F deote their distributio fuctio ad their survivor fuctio 1 F. We itroduce a cotiuous distributio fuctio F c which agrees with F o the itegers ad is defied by liear iterpolatio i log elsewhere. Thus for ay x, 27 x + 1 c x x where c is the survivor fuctio of F c. Sice c is strictly decreasig, we may defie a sequece of costats γ c by the equatio 28 c γ c =1

16 968 C. W. ANDERSON, S. G. COLES AND J. HÜSLER Fig. 4. Effect of differet ormalizatios o the distributio fuctio of the ormalized maximum of idepedet Poisso radom variables with mea λ = log 5/2. Double log scale; = Step fuctio labelled r = 1 based o ormalizig costats asserted by Propositio 1 to be appropriate for this case; fuctio labelled r = 0 based o ormalizatio appropriate for fastergrowig λ. for each positive iteger. It is easy to see that for large the γ c are the approximate e 1 quatiles of the distributio of the maximum of idepedet radom variables with distributio fuctio F c, ad so they provide a approximatio to the locatio of the distributio of max 1 i R i. We first establish a lower boud o the rate of growth of γ c whe λ = o log. From 27 ad 28 we have 29 1 = c γ c γ c + 1 λ γ c +2 γ c + 2! e λ = λg g! e λ say, where g = γ c + 2. O takig logs of 29 ad usig Stirlig s approximatio g! 2π 1/2 e g g g+1/2 we fid that 30 log λ + g log eλ log g 1 2 log g must be bouded above. Necessarily therefore g>eλ evetually. Note that i this regio, 30 is mootoic decreasig i g. To obtai a better boud suppose that g = λ log 1/2 λ log 1/2. Sice λ = o log, such a g

17 MAXIMA OF POISSON-LIKE VARIABLES 969 Fig. 5. The quality of covergece i relatio to growth of λ for x = whe λ = 2log 5/2 is certaily i the regio g>eλ for large eough. So expressio 30 is o less tha 1/2 } log λ + λ log {1 1/2 λ + log 1 log 4 log λ log { log 1 λ 1/2 1/2 log + λ λ log log λ } log log log 4 log which goes to with. Thus it must be true that g> λ log 1/2 for all large eough. However, for ay sequece γ growig i such a way that γ/ λ log 1/2 is bouded away from 0 it is true that 31 lim γ + 1 = 0 γ To prove 31 ote that e λ λ m+1 λ m+1 m + 1! < m <e λ m + 1! 1 λ 1 m + 2

18 970 C. W. ANDERSON, S. G. COLES AND J. HÜSLER for iteger m>λ 2. Thus γ + 1 γ < λ γ λ γ < λ 1 λ 1 γ γ which goes to 0 as icreases. It is easily verified from 31 ad the defiitio of c that for ay positive ε ad for γ/ λ log 1/2 bouded away from 0, c γ + ε 32 lim = 0 c γ I particular 32 holds for γ = γ c ad γ = γ c ε, so that c γ c + ε = c γ c + ε 0 c γ c ad c γ c ε = c γ c ε c γ c as. It follows that lim F c γ c + ε =1 lim F c γ c ε =0 ad so that lim F γ c ε =1 lim F γ c ε =0 Thus we have proved the followig propositio. Propositio 6. that Whe λ = o log, there is a sequece of itegers I such lim P max R i = I or I + 1 = 1 1 i I this case, therefore, max 1 i R i behaves i the same way as whe λ is costat. 7. Discussio. For practical applicatios we require a approximate family for the distributio of max 1 i R i where the R i have ukow distributio. What we have show i this paper is that if the R i are Poisso with mea λ ad we cosider max 1 i R i as a poit o a suitable path of variables of the form max 1 i R i, where the R i are Poisso with mea λ, a Gumbel approximatio is valid. This is supported also by umerical calculatios. As show i Propositios 2, 3 ad 4, our results do ot deped critically o the R i beig Poisso variables; the Gumbel limit for maxima is foud to be valid for the etire class of distributios satisfyig the coditios of Cramér s theorem, ad also for some distributios with heavier tails. This robustess is crucial for statistical applicatios where the paret populatio is ukow.

19 MAXIMA OF POISSON-LIKE VARIABLES 971 Returig specifically to the case of Poisso variables, we obtai also that the Gumbel approximatio for maxima will ot be good if λ is so small relative to that the degeerate limit of Propositio 6 is domiat. I the case of gamma radiatio couts, λ is typically of the order λ 1000 for 10-miute couts, of which there are approximately 53,000 i a year. The relative magitude of these particular values suggests it is etirely reasoable to model aual maxima of such couts usig a extreme value distributio. Ackowledgmets. helpful commets. We are grateful to a referee ad Associate Editor for REFERENCES Aderso, C. W Extreme value theory for a class of discrete distributios, with applicatios to some stochastic processes. J. Appl. Probab Aderso, C. W Local limit theorems for the maxima of discrete radom variables. Math. Proc. Camb. Philos. Soc Christoph, G. ad Wolf, W Covergece Theorems with a Stable Law. Akademie, Berli. Embrechts, P. ad Goldie, C. M O the closure ad factorizatio theorems for subexpoetial ad related distributios. J. Austral. Math. Soc. Ser. A E Garde The Argus Project. Tye ad Wear Emergecy Plaig Newsletter 6 8. Feller, W A Itroductio to Probability Theory ad Its Applicatios 2, 2d ed. Wiley, New York. Fisher, R. A. ad Tippett, L. H. C Limitig forms of the frequecy distributio of the largest ad smallest members of a sample. Proc. Camb. Philos. Soc Gomes, M. I Peultimate behavior of the extremes. I Extreme Value Theory ad Its Applicatios J. Galambos, ed Kluwer, Dordrecht. Kimber, A. C A ote o Poisso maxima. Z. Wahrsch. Verw. Gebiete Leadbetter, M. R., Lidgre, G. ad Rootzé, H Extremes ad Related Properties of Radom Sequeces ad Processes. Spriger, New York. Nagaev, A. V. 1969a. Itegral limit theorems for large deviatios whe Cramér s coditios are violated. Izv. Akad. Nauk. UzSSR Ser. Fiz.-Mat. Nauk I Russia. Nagaev, A. V. 1969b. Itegral limit theorems for large deviatios whe Cramér s coditio is ot fulfilled I, II. Theory Probab. Appl , Nagaev, S Large deviatios of sums of idepedet radom variables. A. Probab Petrov, V. V Sums of Idepedet Radom Variables. Spriger, Berli. C. W. Aderso School ofmathematics ad Statistics Uiversity of Sheffield Sheffield S3 7RH Uited Kigdom c.w.aderso@sheffield.ac.uk S. G. Coles Departmet ofmathematics ad Statistics Uiversity oflacaster Lacaster LA1 4YF Uited Kigdom s.coles@lacaster.ac.uk J. Hüsler Istitut für Mathematische Statistik ud Versicherugslehre Uiversität Bere Sidlerstrasse 5 CH-3012 Bere Switzerlad huesler@math-stat.uibe.ch