A CLUSTERING LAW FOR SOME DISCRETE ORDER STATISTICS

Transcription

1 J App Prob 40, (2003) Printed in Israe Appied Probabiity Trust 2003 A CLUSTERING LAW FOR SOME DISCRETE ORDER STATISTICS SUNDER SETHURAMAN, Iowa State University Abstract Let X 1,X 2,,X n be a sequence ofindependent, identicay distributed positive integer random variabes with distribution function F Anderson (1970) proved a variant ofthe aw ofarge numbers by showing that the sampe maximum moves asymptoticay on two vaues ifand ony iff satisfies a custering condition, [1 F(n+ 1)]/[1 F (n)] 0 In this artice, we generaize Anderson s resut and show that it is robust by proving that, for any r 0, the sampe maximum and other extremes asymptoticay custer on r + 2 vaues ifand ony if [1 F(n+ r + 1)]/[1 F (n)] 0 Together with previous work which considered other asymptotic properties ofthese sampe extremes, a more detaied asymptotic custering structure for discrete order statistics is presented Keywords: Order statistics; aw ofarge numbers; extremes; asymptotics; custering AMS 2000 Subject Cassification: Primary 62G32 Secondary 47F30 1 Introduction and resuts Let X 1,X 2,,X n, be a sequence ofindependent, identicay distributed (iid) reavaued random variabes with common cumuative distribution function F(x) P(X 1 x) < 1 for a x < Let X (1) X (2) X (n) be the order statistics in a sampe of size n The maxima order statistic is X (n) and, for k 0, X (n k) is caed the extreme order statistic The cassica asymptotic theory ofthe sequence X (n) } incudes Gnedenko s necessary and sufficient condition for a aw of arge numbers to hod (see Gnedenko (1943) or Gaambos (1987) for a modern account) Proposition 11 There exists a sequence v n } such that, for a ε>0, P( X (n) v n ε) 0 ifand ony if, for a x>0, 1 F(x + y) 0 (11) y 1 F(y) When the random variabes X n are discrete, say integer vaued, then ofcourse Gnedenko s theorem cannot hod since, for x<1, the it appearing in (11) is 1 However, something can sti be said in this case Received 21 March 2001; revision received 15 May 2002 Posta address: 400 Carver Ha, Department ofmathematics, Iowa State University, Ames, IA 50011, USA Emai address: sethuram@iastateedu Research supported in part by NSF grant DMS

2 A custering aw for some discrete order statistics 227 Henceforth, et us specify that the variabes X 1,X 2, take vaues on the positive integers We define p t P(X 1 t) and set p t 1 F(t 1) j t p j for t 1 We wi assume throughout, uness specified otherwise, that p t > 0 for infinitey many t 1 (so that F(t) < 1 for a t 1) There wi be occasion, however, in Sections 12 and 13, when we wi assume the stronger condition that p t is positive eventuay, namey when F beongs to S F : p t > 0 for a t t 0, some t 0 1} Anderson (1970) proved the foowing resut under the assumption that F S, athough virtuay the same proofhods under our more genera assumption that F(t) < 1 for a t 1 Proposition 12 There exists an integer sequence v n } such that P(X (n) v n or v n + 1) 1 ifand ony if 1 F(n+ 1) 0 (12) 1 F (n) Note that Anderson s condition (12) is Gnedenko s condition (11) ony for x 1, and woud therefore give the weaker resut Anderson s condition (12) can be equivaenty stated as t p t / p t 1, or that the hazard rate converges to 1 In this paper, motivated by a previous investigation ofdiscrete order statistics, we extend Anderson s resut to other extreme order statistics and aso to more genera but reated discrete distributions These extensions carify further the asymptotic structure of the extreme statistics under these distributions in terms ofsome custering interpretations Consider the foowing casses of discrete distributions identified by their iting hazard rate behavior For integers r 0, define } 1 F(n+ r + 1) A r F : 0 1 F (n) Note that Anderson s condition (12) reduces to F A 0 It is easy to see that A r A s for r s (see Lemma 11 beow), and so A r can be thought ofas a generaization ofanderson s distributions The distributions in A r are roughy characterized by ight tais For instance, the Poisson distribution ies in A 0, but the geometric distribution, with positive iting hazard rate, beongs to no A r for r 0 The foowing was stated in the penutimate part of Theorem 13 of Athreya and Sethuraman (2001) (We point out that the proofoftheorem 13 in Athreya and Sethuraman (2001) in fact assumes the condition F S but this is not noted expicity) Proposition 13 Consider a distribution F S Then, for r 0, F A r ifand ony if P( X (n) X (n 1) >r+ 1) 0 This resut suggests a sort of custering ofextreme vaues X (n) and X (n 1) under distribution F A r on sets ofcardinaity r + 2 Our main resuts generaize Proposition 12 and the if part of Proposition 13, and show that a certain custering does hod, not ony for the first two extreme statistics, but for the coections X (n),,x (n k) } for k 0 and go on to specify a set of custering vaues

3 228 S SETHURAMAN Theorem 11 For r 0, there exists an integer sequence v n,r v n,r (F ) for n 1 such that P(X (n k) v n,r,v n,r + 1,,v n,r + r + 1}) 1 for each k 0 ifand ony iff beongs to A r We remark that the sequence v n,r can be taken expicity as v n,r u n,r , where u n,r+1 supz : 1 F r+1 (z) 1/n}, F r+1 is defined in Section 2 and denotes the integer-part function Define V n,r to be the set ofcustering vaues, V n,r v n,r,v n,r + 1,,v n,r + r + 1} An immediate coroary oftheorem 11 which specifies asymptotic custering on V n,r is the foowing Coroary 11 For r 0, there exists an integer sequence v n,r v n,r (F ) for n 1 such that P(X (n),,x (n k) V n,r ) 1 for a k 0 ifand ony iff beongs to A r Given now that the extremes hover on the set V n,r asymptoticay, it is natura to ask how the extreme vaues move inside V n,r With respect to a arge cass ofdistributions in A 0 and A r \ A r 1 for r 1, we show in Section 12 that the extremes move back and forth on the endpoints v n,r and v n,r + r + 1 infinitey often (io) with probabiity 1 (see Proposition 16) With additiona assumptions, we aso show that the extremes move on a points in V n,r infinitey often with probabiity 1 (see Proposition 17) It is interesting to note, on the other hand, that intermediate movement (that is, movement on V n,r \v n,r,v n,r + r + 1}) may not be possibe for some exampes (see Exampe 11) Any more precise determination of the boundaries separating these phenomena is open however We remark now on the pan ofthe paper In Section 11 we make some remarks on the structure ofthe distribution sets A r for r 0 Then, in Section 12, we prove the fine custering properties auded to above In Section 13, we comment on some compements to Theorem 11 Finay, in Section 2, we prove Theorem Casses A r An equivaent characterization ofthe set A r is that A r contains those distributions such that the probabiity ofexpiring by time n + r given that one has ived up to time n tends to 1, 1 r } A r F : p n+ 1 p n As mentioned before, A r oosey characterizes distributions which custer in a certain way on ow vaues or have ight tais As r grows, these tais grow heavier More precisey, there is a natura grading ofthe casses A r Lemma 11 For r 0, A r A r+1 Proof This incusion foows easiy from the identity 1 F(n+ r + 2) 1 F (n) 0 1 F(n+ r + 2) 1 F(n+ 1) 1 F(n+ 1) 1 F (n)

4 A custering aw for some discrete order statistics 229 Later, in Proposition 14, we specify distributions beonging to A r+1 \ A r for r 0 It wi be usefu now to compare and contrast the sets A r with the foowing casses of distributions Define, for r 0, } p n B F : 0, p n } p n+r C r F : 0 p n The set B, in contrast to the sets A r, consists ofdistributions with heavy tais For instance, the zeta distributions beong to B In fact, B and the sets A r for r 0 are disjoint Lemma 12 For each r 0, A r B Proof We have 1 F(n+ r + 1) 1 F (n) r 0 1 F(n+ + 1) 1 F(n+ ) r [ 0 1 p n+ p n+ Then, for a distribution p n } B, we have that (1 F(n+r +1))/(1 F (n)) 1asn Hence, F A r for any r 0 With respect to the sets C r, it is immediate that C 0 B But, on the other hand, C r contains both B and A r 1 for r 1 ] Lemma 13 For r 1, C r B A r 1 Proof Write p n+r p n+r p n+r p n p n+r p n Observe that both factors on the right-hand side are bounded by 1 Moreover, as n, p n+r / p n+r vanishes when p n } B and p n+r / p n vanishes when p n } A r 1 These observations impy the emma In genera, we can construct distributions in C r \ (B A r 1 ) for r 1 However, the sets C r and A r \ A r 1 are disjoint for r 1 Lemma 14 For r 1, C r (A r \ A r 1 ) Proof We have r 1 0 p r0 n+ p n+ p n+r p n p n p n If p n } A r C r, the right-hand side above vanishes as n, and so p n } A r 1 At this point, we specify a cass of cycic distributions which beong to A r \A r 1 for r 1 Let now Z m 1, 2,,m} for m 1

5 230 S SETHURAMAN Proposition 14 Let Y be a random variabe whose distribution function F Y beongs to A 0 Let aso r 1, and suppose that Z n } is a famiy of random variabes on Z r+1, independent of Y, such that inf P(Z n r + 1) >0 Then the aw of X (r + 1)(Y 1) + Z Y beongs to A r \ A r 1 (13) The variabe X has the interpretation that Y fixes a certain eve and Z Y determines, through an (r + 1)-sided die ro, how much more to assign to X We wi ca variabes X satisfying (13) fu (r + 1)-cycic random variabes with eve Y and increment Z Y We point out aso that the fu (r + 1)-cycic variabes incude those distributions where the increment does not depend on the eve, Z n Z In this homogeneous case, X is generated more simpy by eve Y and an independent (r + 1)-sided die ro In addition, we remark that it is tempting to think that a distributions in A r \ A r 1 are generaizations ofthe fu (r + 1)-cycic aws identified in (13) However, the rate at which the hazard rate of Y vanishes seems to be invoved in proving a converse, and the issue is not further pursued here The foowing wi be of hep in proving Proposition 14 Lemma 15 Let p t } A 0 Let aso c 1 (t)} and c 2 (t)} be sequences ofnumbers 0 c 1 (t), c 2 (t) 1 such that inf t c 1 (t) > 0 Then and t t c 1 (t)p t c 1 (t)p t + p t+1 1 (14) c 2 (t)p t+1 c 1 (t)p t + p t+1 0 (15) Proof As p t+1 p t p t, we may rewrite the denominator in (14) as [ c 1 (t)p t + p t+1 p t 1 + (c 1 (t) 1) p ] t p t But aso, since p n p n p n+1 p n (1 p n / p n ), we can write the denominator in (15) as [ c 1 (t)p t + p t+1 p t c 1 (t) p ] t p t+1 p t+1 [1 + c 1 (t) p t p t (1 p t / p t ) From these cacuations, as t p t / p t 1, the emma foows straightforwardy We now prove Proposition 14 for r 1 A generaized argument for r 1 is sketched in Appendix A ProofofProposition 14 for r 1 Let p t } A 0 be the distribution of Y and et (α t, 1 α t ) be the distribution of Z t on Z 2 where inf t 1 α t > 0 and so sup t α t < 1 Then the distribution of X is the distribution q t } given by α t p (t+1)/2 for t odd, q t (1 α t )p t/2 for t even ]

6 A custering aw for some discrete order statistics 231 Then We first show that X A 0 Indeed, we can compute that q t q t + q t+1 + α t p (t+1)/2 + (1 α t )p (t+1)/2 + p (t+1)/2 for t odd, (1 α t )p t/2 + αp (t+2)/2 + (1 α t )p t/2 + p (t+2)/2 for t even q t q t α t p (t+1)/2 p (t+1)/2 for t odd, (1 α t )p t/2 for t even (1 α t )p t/2 + p t/2+1 As p n / p n 1, we concude that, for t odd, sup t q 2t 1 / q 2t 1 < 1, but that, for t even, t q 2t / q 2t 1 using (14) in Lemma 15 Therefore, q t } A 0 However, X does beong to A 1 Indeed, q t + q t+1 q t p (t+1)/2 p (t+1)/2 for t odd, (1 α t )p t/2 α t p (t+2)/2 + for t even (1 α t )p t/2 + p (t+2)/2 (1 α t )p t/2 + p (t+2)/2 Ceary, for t odd, the fraction (q t + q t+1 )/ q t converges to 1 But, aso for t even, from Lemma 15, this fraction tends to 1 Hence, t (q t + q t+1 )/ q t 1 and so q t } A 1 12 Fine custering We investigate the custering interpretation ofcoroary 11 further A natura starting point in this regard is to ask about the asymptotic behavior ofthe extremes among the custering vaues Do they move together or separate themseves from each other with high probabiity? The answer to the ast query is in the negative on both hypotheses, and is addressed in Propositions 16 and 17 and Exampe 11 Consider the foowing resut from the first part of Theorem 13 in Athreya and Sethuraman (2001) Proposition 15 Consider a distribution F S (i) For each 1, F B ifand ony ifthe it P(X (n) X (n 1) X (n +1) >X (n ) ) exists (ii) For each m 1, F B A m 1 ifand ony ifthe it P(X (n) >X (n 1) + m) exists (iii) For each m 1, F C m ifand ony ifthe it P(X (n) X (n 1) X (n +1) >X (n ) + m) exists for each 2

7 232 S SETHURAMAN A consequence ofthis proposition which iuminates fine custering properties ofthe extremes is the foowing To simpify notation, define, for r 0, A 0 for r 0, D r A r \ A r 1 for r 1 Proposition 16 Consider a distribution F D r S for r 0 Then, for each k 1 and 1 k, P(X (n) X (n +1) v n,r + r + 1 and X (n ) X (n k) v n,r io) 1 and P(X (n) X (n 1) X (n k) v n,r io) 1 Proof For r 0, if F D r, then necessariy F B by Lemma 12 Aso, for r 1, if F D r, then we must have F C r by Lemma 14, and in particuar F B A r 1 by Lemma 13 Then, by Proposition 15, when F D r, the it P(X (n) X (n +1) >X (n ) + r) (16) does not exist for 1 However, by Coroary 11, P(X (n),x (n k) V n,r ) 0, (17) where ofcourse the diameter ofv n,r is r + 1 Hence, using (17) to decompose (16), we have that P(E n,,k,r ) does not exist where, for n k 1, E n,,k,r X (n) X (n +1) v n,r + r + 1 and X (n ) X (n k) v n,r } Therefore, P(E n,,k,r io) sup P(E n,,k,r )>0 (18) Simiary, by Theorem 11, as F A r 1, we have that P(X (n) v n,r +1,,v n,r +r +1}) 1 But, as F A r,wehavep(x (n) V n,r ) 1 from the positive part of Theorem 11 Therefore, P(X (n) v n,r ) 0 Hence, as P(X (n) X (n k) V n,r ) 1, by Coroary 11 we have that P(X (n) X (n k) v n,r io) sup P(X (n) X (n k) v n,r )>0 (19) We now argue that the infinitey often events in (18) and (19) are with fu measure For each j 0, et G j X (n j) } Then, as P(G j ) 1forj 0, the events kj0 G j X (n) X (n k) v n,r io} and k j0 G j E n,,k,r io} beong to the tai σ -fied ofthe iid observations X i } Then, by Komogorov s 0 1 aw and some set-agebra, we have that P(X (n) X (n k) v n,r io) P(E n,,k,r io) 1 The resut indicates that the sampe extremes X (n),x (n 1),,X (n k), under distributions F D r S, behave quite irreguary, moving on top ofeach other and as far as the maximum of r + 1 units away from each other infinitey often What is not mentioned in the proposition is ifthe extremes move to intermediate distances reative to each other infinitey often It is interesting to reaize that such intermediate movement may not be possibe

8 A custering aw for some discrete order statistics 233 Exampe 11 Let p t } A 0, and consider the distribution q t } given by (1 α t/2 )p t/2 for t even, q t α (t+1)/2 p (t+1)/2 for t odd, where 0 α t < 1 and t α t < is a summabe sequence ofnumbers By Proposition 14 for r 1, we have that q t } A 1 \ A 0 Moreover, the odd and even probabiities q 2t 1 and q 2t can be interpreted as the chance ofchoosing t under p t } and then obtaining heads or tais respectivey in an independent coin-toss Z t with P(Z t H) α t and P(Z t T) 1 α t Then, for the iid sequence X i } under q t }, for each 0, as X (n ) amost surey, P(X (n ) is odd io) P(X (n ),X (n ) is odd io) P(Z t H io) 0 In particuar, the extreme vaues X (n),,x (n k) take even vaues eventuay with probabiity 1 So, amost surey, the reative intermediate distance of1 is not achieved infinitey often by these extremes On the other hand, we might beieve, under some natura assumptions, that the extremes under distributions F X A r \ A r 1 for r 1 considered in Proposition 14 shoud achieve a possibe reative intermediate distances between 0 and r + 1 infinitey amost surey This is indeed the case Let k, r 0, and define a set ofordered vectors, C(k,r) x x 0,,x k :r + 1 x 0 x 1 x k 0} Define aso for x C(k,r) the shifted vector x n,r x 0 + v n,r,,x k + v n,r V k+1 n,r Proposition 17 Let r 1, and et X be a fu (r + 1)-cycic variabe with eve Y and increment Z Y satisfying (13) Let F X A r \ A r 1 and F Y A 0 be the distributions of X and Y respectivey Suppose aso that F Y S and that Z n } satisfies the condition Then Aso, for k 0, when x C(k,r), inf P(Z n ) > 0 for a Z r+1 v n,r (F X ) (r + 1)v n,0 (F Y ) P( X (n),x (n 1),,X (n k) x n,r io) 1 Proof Let u 1 and et Y (n),,y (n u) } be the first u + 1 sampe extremes under F Y in a sampe ofsize n>u From (18) appied to the Y i } sampe, we have for 1 m u that sup P(E n,m,u,0 )>0, where E n,m,u,0 Y (n) Y (n m+1) Y (n m) + 1 Y (n u) + 1 v n,0 + 1} On the event E n,m,u,0, there are exacty m variabes which take the maximum vaue v n,0 + 1, and at east u m + 1 variabes exacty one unit ess in the Y i } sampe Moreover, as the Z n } are Z r+1 -vaued variabes, we have X (n),,x (n m+1) } (r + 1)v n,0 + Z r+1 and X (n m),,x (n u) } (r + 1)(v n,0 1) + Z r+1 ifand ony ife n,m,u,0 occurs

9 234 S SETHURAMAN Specify now increments 0,, m 1 with r m 1 1 As the Z Yi } variabes are independent and Y (n j) amost surey for j 0, sup P(X (n) (r + 1)v n,0 + 0,,X (n m+1) (r + 1)v n,0 + m 1, X (n m),,x (n u) } (r + 1)(v n,0 1) + Z r+1 ) m 1 sup P(E n,m,u,0 ) inf P(Z n i )>0 (110) i0 since inf P(Z n ) > 0 for a Z r+1 by assumption Let us choose now 0 r +1 With this choice, as P( X (n) X (n u) >r+1) 0 from Coroary 11, we obtain further that sup P(X (n) (r + 1)(v n,0 + 1), X (n 1) (r + 1)v n,0 + 1,,X (n m+1) (r + 1)v n,0 + m 1, X (n m) X (n m q) (r + 1)v n,0 )>0 Moreover, as the stronger caim, P(X (n),,x (n u) V n,r ) 1, aso hods from Coroary 11, we must have sup P(X (n) (r + 1)(v n,0 + 1), X (n 1) (r + 1)v n,0 + 1,,X (n m+1) (r + 1)v n,0 + m 1,X (n m) X (n u) (r + 1)v n,0, X (n) v n,r + r + 1,X (n m) X (n u) v n,r )>0 This yieds (r + 1)v n,0 v n,r Substituting this reation into (110) and noting once more that P(X (n),,x (n u) V n,r ) 1 (from Coroary 11) gives that sup P(X (n) v n,r + 0,X (n 1) v n,r + 1,,X (n m+1) v n,r + m 1, X (n m) X (n u) v n,r )>0 (111) On the other hand, by (19), as the assumptions on Y and Z n } impy that F X (A r \A r 1 ) S, we have that sup P(X (n) X (n u) v n,r )>0 (112) We now associate to each vector x C(k,r) one ofthe above its When x is such that x k > 0, we choose (111) with m u k + 1 and x i i for 0 i k However, when x is such that x i 1 > 0 and x i x k 0fori k, we choose (111) with m i, u k, and x j j for 0 j i 1 Finay, when x is such that x 0 x k 0, we choose (112) Taking account ofthese choices, we have, for each x C(k,r), that P((X (n),,x (n k) ) x n,r io) sup P((X (n),,x (n k) ) x n,r )>0 The argument now is anaogous to the part ofthe proofofproposition 16 foowing (19) We eave open, however, any more carefu investigation of the boundaries in A r \ A r 1 for r 1 which separate distributions eading to the two extreme vaue behaviors mentioned in Exampe 11 and Proposition 17

10 A custering aw for some discrete order statistics Compements We comment now on the asymptotic behavior ofsampe extremes when the underying distribution F is not in A r for any r 0, the contrapositive oftheorem 11 In some sense, the antithesis ofcustering occurs It turns out that the sampe maximum, X (n), is asymptoticay distinct from the other extremes and strongy so The resuts are most compete when F is a priori restricted to S We ist some ofthese resuts beow, focusing on the cass B Anderson (1970) showed that no aw ofarge numbers is possibe for the maxima statistic since P( X (n) v n <y) 0 for any sequence v n } and any y>0ifand ony if F B Noting Proposition 21 beow, the same statement can be made to hod for extremes X (n k) for k 1 Baryshnikov et a (1995) considered the asymptotic chance ofa draw and proved that the it P(X (n) >X (n 1) ) exists ifand ony iff B, and that it equas 1 when it exists Athreya and Sethuraman (2001) showed that the maximum is strongy separated asymptoticay from the other vaues, P(X (n) >X (n 1) + m) 1 for any m 0, and no type ofdraw is possibe with the maximum, P(X (n) X (n k+1) >X (n k) ) 0for any k 2, ifand ony iff B, among other resuts 2 Proof of Theorem 11 The proof of Theorem 11 foows from a generaization of the method of Anderson (1970) The strategy is in two steps Step 1 We map each distribution function F A r to a continuous distribution function, F r say, which satisfies the assumptions ofgnedenko s aw ofarge numbers Step 2 We then unrave the mapping to get the desired statement for F There is no probem in finding a map for Step 1 The difficuty, however, is in choosing this map so that Step 2 can be managed The contribution here is in finding such a map which generaizes Anderson s for the case r 0 Let m 1 be fixed and, for x>0and 0 m 1, define x m, to be the argest integer ofthe form km + ess than x for k 0 Ofcourse, x x 1,0 represents as usua the greatest integer ess than x Let F be a discrete distribution function on the positive integers For each n 1, et h(n) og(1 F (n)) Note that h is an increasing function on the positive integers and h(n) Define now h m, to be an interpoated version of h: forx 0, h m, (x) h( x m, ) + x x m, (h( x m, + m) h( x m, )) m Note that h m, is continuous, increasing, and aso that x h m, (x) Define now, for x m, the distribution function F m, by its tai, 1 F m, (x) e hm,(x) (the definition for x < m is not reevant for what foows) Define aso the distribution function F m by F m (x) maxf m,0 (x),,f m,m 1 (x)} Some simpe properties seen from the definitions are F m, ( x m, ) F( x m, ) and, as x m, x< x m, + m, 1 F( x m, + m) 1 F m, (x) 1 F( x m, )

11 236 S SETHURAMAN Aso, as x x m,0 for some 0 0 m 1, we have F(x) F( x ) F m,0 ( x m,0 ) F m ( x ) F m (x) (21) It wi turn out that this ast inequaity is the key to the unraveing in the second step mentioned above Let us aso define the sequence u n,m } n 1 for m 1by u n,m sup z : 1 F m (z) 1 } n As F m is continuous, 1 F m (u n,m ) 1/n Aso, as F m (z) < 1forz<, we have, for fixed m, that u n,m increases to infinity as n goes to infinity The foowing two emmas estabish Step 1 Lemma 21 Suppose that F beongs to A r for r 0 Then, for 0 r and 0 <y< 2 1, 1 F r+1, (x + y) 0 x 1 F r+1, (x) Proof By assumption, we have that [1 F(n+ r + 1)]/[1 F (n)] vanishes as n tends to infinity This impies that [h(n + r + 1) h(n)] We now show that [h r+1,(x + y) h r+1, (x)], x from which Lemma 21 easiy foows We have h r+1, (x + y) h r+1, (x) [h( x + y r+1, ) h( x r+1, )] + x + y x + y r+1, (h( x + y r+1, + r + 1) h( x + y r+1, )) r + 1 x x r+1, (h( x r+1, + r + 1) h( x r+1, )) (22) r + 1 Consider now the cases when (i) r + 1 y x x r+1, r + 1 y/10, (ii) x x r+1, >r+ 1 y/10, (iii) x x r+1, <r+ 1 y (i) Here, x + y r+1, x r+1, + r + 1 and the right-hand side of(22) becomes [h( x r+1, + r + 1) h( x r+1, )] + x x r+1, r 1 + y (h( x r+1, + 2r + 2) h( x r+1, + r + 1)) r + 1 x x r+1, (h( x r+1, + r + 1) h( x r+1, )) r + 1 y 10(r + 1) [h( x r+1, + r + 1) h( x r+1, )]

12 A custering aw for some discrete order statistics 237 (ii) Here, aso x + y r+1, x r+1, + r + 1 and the right-hand side of(22) becomes [h( x r+1, + r + 1) h( x r+1, )] + x x r+1, r 1 + y (h( x r+1, + 2r + 2) h( x r+1, + r + 1)) r + 1 x x r+1, (h( x r+1, + r + 1) h( x r+1, )) r + 1 9y 10(r + 1) [h( x r+1, + 2r + 2) h( x r+1, + r + 1)] (iii) Here, x + y r+1, x r+1, and the right-hand side of(22) becomes y r + 1 [h( x r+1, + r + 1) h( x r+1, )] Putting (i) (iii) together gives that (22) is greater than y 10(r + 1) minh( x r+1, +r +1) h( x r+1, ), h( x r+1, +2r +2) h( x r+1, +r +1)}, which diverges with x Lemma 22 Suppose that F beongs to A r for r 0 Then, for any y with 0 <y< 1 2, 1 F r+1 (x + y) 0 x 1 F r+1 (x) Proof Observe that, for fixed x,1 F r+1 (x) 1 F r+1,0 (x) for some 0 with 0 0 r, and so 1 F r+1 (x + y) 1 F r+1 (x) The proof now foows from Lemma 21 1 F r+1(x + y) 1 F r+1,0 (x) 1 F r+1, 0 (x + y) 1 F r+1,0 (x) max 1 Fr+1, (x + y) 1 F r+1, (x) The foowing resut wi be usefu in showing that the maximum and other extremes a have the same behavior Proposition 21 Let x n be a sequence ofnumbers tending to infinity, and et ε>0be fixed Then P( X (n) x n <ε) 1 ifand ony if P( X (n k) x n <ε) 1 for a k 0 }

13 238 S SETHURAMAN Proof We first reca a standard way to determine the distribution function of the sampe extremes Let N n (x) n i1 1 [Xi >x] Then the distribution of X (n k) for k 0 satisfies P(X (n k) x) 1 P(N n (x) k + 1) k ( ) n (1 F(x)) i F n i (x) i i0 Let now x n be a sequence tending to infinity The proofnow foows once we show that F n (x n ) 1 ifand ony if P(X (n k) x n ) 1 and F n (x n ) 0 ifand ony if P(X (n k) x n ) 0 Observe now that F n (x n ) tends to 1 or 0 ifand ony ifn(1 F(x n )) tends to 0 or respectivey Correspondingy, P(X (n k) x n ) tends to 1 or 0 ifand ony if og P(X (n k) x n ) tends to 0 or respectivey We have og P(X (n k) x n ) [ nog F(x n ) og 1 + k [( n )] ] n i i (1 F(x n )) i F i (x n ) i1 [ n(1 F(x n ))(1 + o(1)) og 1 + n i k i1 ] 1 i! (n(1 F(x n))) i (1 + o(1)) as n tends to infinity In the it, the og term is subordinate to the first term This proves the proposition We can now prove Theorem 11 ProofofTheorem 11 We prove the resut for the maximum X (n) The resut for the other extremes woud then foow from Proposition 21 Sufficiency From the definition of u n,r+1, observe that, for 0 <ε< 1 2, n(1 F r+1 (u n,r+1 + ε)) 1 F r+1(u n,r+1 + ε) 1 F r+1 (u n,r+1 ) and 1 n(1 F r+1 (u n,r+1 ε)) 1 F r+1(u n,r+1 ) 1 F r+1 (u n,r+1 ε) Both right-hand sides vanish as n tends to infinity from Lemma 21 Therefore, we concude that F n r+1 (u n,r+1 + ε) 1 and F n r+1 (u n,r+1 ε) 0 (23)

14 A custering aw for some discrete order statistics 239 Now observe from (21) that F(u n,r+1 ε) F r+1 (u n,r+1 ε) and F r+1 (u n,r+1 + ε) maxf r+1, (u n,r+1 + ε)} maxf r+1, ( u n,r+1 + ε r+1, + r + 1)} maxf( u n,r+1 + ε r+1, + r + 1)} F( u n,r+1 + ε +r + 1) F(u n,r+1 + ε + r + 1) From (23), ( P X (n) V n < r + 1 ) + ε 1 2 with V n u n,r+1 + (r + 1)/2 As the maximum takes ony integer vaues, sufficiency in the theorem is proved with v n,r u n,r Necessity Define D n by v n,r + r + 1 for r even, 2 D n v n,r + r for r odd 2 4 For ε>0, we then have ( P X (n) D n r + 1 ) + ε 1 f or r even 2 and ( P r + 1 ε X (n) D n r ε + 1 ) 1 f or r odd 2 4 Let us concentrate on the even case for the moment In this case, ( F n D n + r + 1 ) ( + ε 1 and 2 F n D n r ) ε 0 From the first it, we see that D n must diverge to infinity, and therefore ( ( n 1 F D n + r + 1 )) ( ( + ε 0 and 2 n 1 F D n r + 1 )) ε 2 We may take D n } to be monotonicay increasing, and so, for any integer a 1, we may find n such that D n 1 a D n Therefore, 1 F(a (r + 1)/2 ε) 1 F(a + (r + 1)/2 + ε) 1 F(D n (r + 1)/2 ε) 1 F(D n 1 + (r + 1)/2 + ε), which diverges to infinity as n tends to infinity Hence, substituting ε 4 1 into the eft-hand side, we have 1 F(a r/2 1) a 1 F(a + r/2) But this ast statement is the same as saying that F beongs to A r

15 240 S SETHURAMAN For r odd, we foow the same steps as for r even to arrive at the statement that 1 F(a (r + 1)/2 + 1/4 ε) 1 F(a + (r + 1)/2 + 1/4 + ε) 1 F(D n (r + 1)/2 + 1/4 ε) 1 F(D n 1 + (r + 1)/2 + 1/4 + ε) diverges to as n Substituting ε 1 8 into the eft-hand side, we have that This is equivaent to F beonging to A r 1 F(a (r + 1)/2) a 1 F(a + (r + 1)/2) 0 Appendix A Proposition 14 for r 1 For the reader s convenience, we give a sketch proofofproposition 14 for r 1 Let p t } and q t } be the distributions of Y and X respectivey Let (αt 1,,αr+1 t ) be the distribution of Z t where 0 αt 1 for 0 r+1, r+1 0 α t 1, and inf t αt r+1 > 0 Let aso S 0,r + 1, 2(r + 1),} Then αt 1p (t+r)/(r+1) for t S + 1, q t αt r+1 p t/(r+1) for t S + r + 1, p (t+r)/(r+1) for t S + 1, ( r+1 ) αt p (t+r 1)/(r+1) + p (t+2r)/(r+1) for t S + 2, q t 2 αt r+1 p t/(r+1) + p (t+r+1)/(r+1) for t S + r + 1 and r 1 q t+ 0 ( r αt 1 ( r+1 αt 2 ( r+1 αt 3 α r+1 ) p (t+r)/(r+1) for t S + 1, ) p (t+r 1)/(r+1) for t S + 2, ) p (t+r 2)/(r+1) + α 1 t p (t+2r 1)/(r+1) for t S + 3, t p t/(r+1) + ( r 1 αt 1 ) p (t+r+1)/(r+1) for t S + r + 1

16 A custering aw for some discrete order statistics 241 We now argue that q t } A r 1 Wehave ( r 1 αt )p (t+r)/(r+1) for t S + 1, p (t+r)/(r+1) r 1 0 q t+ q t ( r+1 2 α t )p (t+r 1)/(r+1) ( for t S + 2, r+1 2 α t )p (t+r 1)/(r+1) + p (t+2r)/(r+1) α r+1 t p t/(r+1) + ( r 1 1 α t )p (t+r+1)/(r+1) α r+1 t p t/(r+1) + p (t+r+1)/(r+1) for t S + r + 1 Then, sup t S+1,t ( r 1 0 q t+)/ q t sup r1 t αt 1 inf t αt r+1 < 1 However, on the other hand, by Lemma 15, sup t S+k,t ( r 1 0 q t+)/ q t 1 for 2 k r + 1 as inf r+1 t k α t inf t αt r+1 > 0 Therefore q t } t 1 A r But q t } A r Indeed, r q t+ 0 p (t+r)/(r+1) for t S + 1, ) p (t+r 1)/(r+1) + αt 1 p (t+2r)/(r+1) for t S + 2, ( r+1 αt 2 ( r t p t/(r+1) + α r+1 1 α t ) p (t+r+1)/(r+1) for t S + r + 1 Then, using Lemma 15, it is not difficut to concude that t ( r 0 q t+ )/ q t 1, meaning that the distribution of X beongs to A r Acknowedgement I thank the referee for suggestions which ed to improvements in the exposition of the artice References Anderson, C W (1970) Extreme vaue theory for a cass of discrete distributions with appications J App Prob 7, Athreya, J S and Sethuraman, S (2001) On the asymptotics ofdiscrete order statistics Statist Prob Lett 54, Baryshnikov, Y, Eisenberg, B and Stenge, G (1995) A necessary and sufficient condition for the existence of the iting probabiity ofa tie for first pace Statist Prob Lett 23, Gaambos, J (1987) The Asymptotic Theory ofextreme Order Statistics Krieger, Mebourne, FL Gnedenko, B V (1943) Sur a distribution ite du terme maximum d une série aéatoire Ann Math 44,