1 Introduction and main results

Transcription

1 The Spacgs of Record Values Tefeg Jag ad Dag L 2 Ocober 0, 2006 Absrac. Le {R ; } be he sequece of record values geeraed from a sequece of..d. couous radom varables, wh dsrbuo from he expoeal famly. I hs paper we sudy he behavor of k-spacgs of R, ha s, R k+ R for. We show ha, uder cera codos, a ormalzed k-spacgs emprcally coverges o a Gamma dsrbuo. Couer examples show ha he resul s o vald whe he codos are volaed. A srog law ad a lmg dsrbuo of he larges ormalzed spacgs are also derved. I parcular, hese resuls coclude ha he k-spacgs go o fy whe he populao dsrbuo has heavy al; he spacgs go o zero whe he al s o heavy. Exac speeds of such covergece are obaed. Iroduco ad ma resuls The record values was frs suded by Chadler 952. For full accou of hsory ad refereces, see [], [7] ad [2]. A lo of properes o record values were udersood: he jo dsrbuos of record values was characerzed, see, e.g., p.65 [2]; he lmg dsrbuos of records were proved o be he exreme value dsrbuos, see, e.g., p. 74 [2]; he exremal process, whch s closely relaed o record values, was also suded, see p. 79 [2], Deheuvels[8] ad [9], ad leraures here; some lm heorems abou record values are suded Bose e al[4] ad [5]. I hs paper, we wll vesgae he spacgs of record values, whch o our kowledge has o bee suded before. The sudy s spred by smlar research radom marx heores see, e.g., Gaud[4], Meha[20], radom graphs see, e.g., Jacobso e al[6], quaum chaos see, e.g., Rudck e al[22], ad umber heores see, e.g., Llewood ad Hardy[5], ad Schmd[23]. The ma cocer hs dreco s he k-spacgs of a gve ragular array of radom or o-radom varables: a, a,2 a,m as s large, where m depeds o. By k-spacgs of hs sequece we mea a,+k a, for =, 2,, m k. Two of he ypcal quesos are he behavor of he emprcal dsrbuos of spacgs ad he larges spacgs as s large. Now we sae our ma resuls hs paper. Suppored par by NSF #DMS ad NSF #DMS , School of Sascs, Uversy of Mesoa, 224 Church Sree, MN55455, jag@sa.um.edu. 2 School of Mahemacs, Jl Uversy, 0 Qawe road, Chagchu3002, P. R. Cha, ldagx@gmal.com. Key Words: record value, exreme value, emprcal dsrbuo, lmg dsrbuo, Se s Posso approxmao mehod. AMS 2000 subjec classfcaos: 60F05, 60F5, 62E20, 60G07.

2 Le X, X, X 2, be..d. radom varables wh cumulave dsrbuo fuco F x ad desy fuco px. Tha s, F x = P X x = x p d. for ay x R. Se L =, L = f{k > L ; X k > X L }, 2..2 The L, L 2, are called upper record mes, ad X L, X L2, X L, are called upper record values. If he par X k > X L.2 s replaced by X k < X L, he he correspodg L s ad X L s are referred o as lower record mes ad lower record values, respecvely. Sce oe egave sg wll chage he lower case o he upper case, we wll cosder he upper case oly hs paper. For coveece of oao, le R = X L for. We wll use he followg codo laer: There exss A [, + such ha px > 0 for x > A, ad px = 0 for x < A ad px s couous o A, +..3 Uder hs codo, we have ha F x = px for all x > A, ad ha he verse of F x exss o A, +. Defe gx = pf x, 0 < x <. The followg codo wll also be used: There exs cosas α > 0 ad β > 0, ad a fuco ωx defed o 0, + wh lm x + ωx = 0 such ha g = β log α /α { + ω }.4 as 0 +. Throughou hs paper, log x = log e x for x > 0. Ths codo looks a b srage a frs sgh. Proposo A. Appedx says ha.4 holds f he desy fuco of radom varable X s of he form px = c e κ x α +bx I{x > A}, for some posve cosas c, κ ad α, ad some cosa A, ad some fuco bx. Whe bx 0, we derve ha β = ακ /α ad ωx = Olog log x/ log x as x + ; f bx s roughly of order ox α / log x as x + see he exac codos Proposo A., we oba ha β = ακ /α ad ωx = o/ log log x as x +. Ths says ha codo.4 holds for a grea class of dsrbuos he expoeal famly. The followg resul shows ha he emprcal dsrbuo of k-spacgs of record values, suably ormalzed, goes o a Gamma dsrbuo. THEOREM Suppose px = ce xα Ix > 0 for some cosas c > 0 ad α > 0, or more geerally, codos.3 ad.4 hold wh ωx = o/ loglog x as x +. Gve eger k. Le D = α /α R k+ R for ad µ,k = / = δ D. The, µ,k coverges dsrbuo o µ wh desy hx = β k x k e βx Ix 0/k! almos surely. 2

3 REMARK. Suppose X has he log-ormal dsrbuo, ha s, log X N0,. The px does o sasfy codo.4 because of Proposo A.2 Appedx, ye he al of hs dsrbuo s faser ha he al wh he form of a raoal fuco, for example, px = x 2 Ix. Proposo A.3 says ha Theorem does o hold whe X has log-ormal dsrbuo. However, he proposo ells ha a dffere formulao of spacgs sll gves a smlar lmg dsrbuo: / = δ D coverges o he Gamma dsrbuo wh desy px = 2 k x k e 2x Ix 0/k!, where D = /2log R k+ log R for. Ths says ha he spacgs amog records become larger ha hose as Theorem. Aoher example s F x = log x Ix e. The F x = exp/ x for x 0,. Oe ca check from.5 below ha LR, R 2,, R = Le es, e es 2,, e es. where S = ξ + ξ ξ for ad ξ s are..d. radom varables wh dsrbuo Exp. The spacgs are eve much larger ha hose as he log-ormal case. REMARK.2 Theorem roughly says ha he k-spacgs of record values mulpled by α /α are radom varables from a Gamma dsrbuo. Recall ha a ypcal dsrbuo of X sasfyg codo.4 s px = ce xα I{x > 0} for some c > 0. If α =, he k-spacgs whou ay ormalzao ca be hough as a radom sample from a Gamma dsrbuo. If α <, he α /α 0, whch meas ha he spacgs whou ormalzao become larger ad larger ad go o fy. If α >, he α /α +. Ths ells us ha he spacgs go o zero. However, sce he order of he -h spacg s / /α, he fe sum of hese spacgs s / /α =. O he oher had, ha s obvous because he sum of he frs spacgs s equal o = R k+ R = +k =+ R k = R R +k k = R. The radom varable k = R does o deped o, ad R +k supposedly goes o + because he rgh ed of he suppor of radom varable X s + from codo.3. The ex resul gves he scale of he larges ormalzed spacgs. THEOREM 2 Suppose px = ce xα Ix > 0 for some cosas c > 0 ad α > 0, or more geerally, codos.3 ad.4 hold wh ωx = O/ loglog x as x +. Gve eger k, le W = max { α /α R k+ R }. The lm W log β Wh he srog law above, wha follows we ge a refed resul abou W, whch s he lmg dsrbuo of he larges spacgs. THEOREM 3 Suppose px = ce xα Ix > 0 for some cosas c > 0 ad α > 0, or more geerally, codos.3 ad.4 hold wh ωx = o/ loglog x as x +. Gve eger k, le W = max { α /α R k+ R }. The, a.s. P W β log β k loglog x exp 3 k! e βx

4 for ay x R. The rgh had sde above s a exreme value dsrbuo. The proofs of he above heorems rely o he followg represeao formula see, e.g., Proposo 4. from [2]: LR, R 2,, R = LF e S, F e S2,, F e S.5 where F s he dsrbuo of X as., ad S k = ξ + + ξ k for k, ad ξ, ξ 2, are..d.radom varables wh dsrbuo Exp. The oule of hs paper as follows. We prese all he proofs of heorems saed above Seco 2; some echcal lemmas used Seco 2 ad some ools o large devaos ad Se s Posso approxmao mehod are provded Seco 3. 2 Proofs I hs seco, we provde he proofs of resuls saed Seco. equvale form of codo.4, whch s covee dscusso laer. We wll use he followg There exs cosas α > 0 ad β > 0, ad a fuco ωx defed o 0, + wh lm x + ωx = 0 such ha as 0 +. R g log α /α β = ω Proof of Theorem. By Theorem.3.3 of Dudley [2], s eough o show ha R fxµ,kdx fxµ dx as + for ay Lpschz fuco fx wh Lpschz orm equal o, parcular, f := sup x R fx. Now R fxµ,k dx = f = α /α R k+ R. We have o show ha he rgh sde above almos surely goes o k! 0 fxβ k x k e βx dx = Sce fx s bouded, we oly eed o prove ha =[c] f k! α /α R k+ R c k! 0 0 f f x x k e x dx. β 2. x x k e x dx 2.2 β for ay c 0,. Recall formula.5, whe we dscuss quaes o {R ; } for fxed, we smply regard R = F e S for. By he Mea-value heorem, here exss [S, S k+ ] such ha e θ α /α R k+ R = S k+ S g e θ α /α

5 Possbly here are may s sasfyg he above equao. For defeess, choose o be he fmum of hose o sasfy he above. By he couy of px as assumed codo.3, he equao 2.3 sll holds for he ew. The such may o ecessarly be measurable. Bu we do o eed ha, he codo S S k+ s eough laer proofs. Defe The, by 2.3, Hx = βx logx α /α, x > g x α /α α /α R k+ R = β S k+ S He θ. 2.5 By codo 2., here exss δ 0, such ha Hx 2β ωx for all 0 < x < δ. From 2.5 ad he Lpschz propery of fx, we have ha f α /α R k+ R f β S k+ S α /α β S k+ S He θ. Obvously, S k+ S k max jk+ ξ j for all. To prove 2.2, suffces o show ha ad =[c] f β S k+ S cef β S k max ξ max +k [c] = c k! 0 f β x x k e x dx a.s. 2.6 α /α He θ 0 a.s. 2.7 To prove 2.6, s eough o show ha fs k+ S EfS k 0 a.s. =[c] Sce =[c] a = = a [c] = a + a [c] for ay {a ; }, he above s equvale o ha [c] U := fs k+ S EfS k 0 a.s. = for ay c 0, ]. Se S 0 = 0. Wre U = k Z,j fs k EfS k Z,j = j=0 [[c] j/k]+ = where {fs k+j S k+j EfS k }. 5

6 Nog ha he radom varables he secod sum are bouded..d. radom varables, s easy o check ha EZ,j 4 = O 2 as for 0 j k. The, sce f, by he covexy of fuco rx = x 4, as. Therefore, EU 4 8E k j=0 Z,j k3 k 4 EZ,j 4 = O 2 P U ɛ ɛ 4 = j=0 = EU 4 <. By he Borel-Caell lemma, U 0 a.s. Thus 2.6 follows. Now P max ξ 3 log P ξ > 3 log = / 2. The = P max ξ 3 log <. By he Borel-Caell lemma aga, lm sup max ξ log 3 a.s. 2.8 Noe ha ab a b + a + b for ay a, b R. Thus, o prove 2.7, suffces o show ha log max 0 a.s. 2.9 [c] He θ α /α log max 0 a.s. 2.0 [c] Gve x > 0 such ha x < /2. By he Mea-value heorem, here exss ξ bewee ad x such ha x α /α = x ξ /α α α 2/α α α x. 2. provded x < /2. Thus, o prove 2.0 s eough o show log max [c] 0 a.s. 2.2 Frs, sce S S k+, max [c] max S [c] [c] max S k + S. [c] [c]+k By he Kolmogorov s srog Law of Large Numbers, S [c] / c a.s. as. Therefore, o see 2.2, s eough o prove ha log max [c]+k S 0 a.s. Ths s obvous, sce 6

7 lm sup S / 2 log log < a.s. by Harma-Wer s Law of Ieraed Logarhm, he max [c]+k S / log log 3/2 log 3/2 max [c]+k sup [c] S 2 log log S 2 log log 0 a.s. as sce he supremum goes o a.s. We oba 2.2. Fally, le s prove 2.9. By codo 2. ad he codo ha ωx = o/ log log x as x +, for ay ɛ > 0, here exss a cosa δ > 0 such ha β log α /α ɛ g log log as 0 < < δ. By he Law of Large Numbers, lm + S = + a.s., herefore max [c] e θ e S [c] 0 a.s. as +. Recallg 2.4, we have ha, wh probably oe, log max ɛ log ɛ a.s. [c] He θ log S [c] as + by he Law of Large Numbers aga. The 2.2 follows. We eed he followg lemma o prove Theorem 3. LEMMA 2. Suppose he codos Theorem 3 hold. Recall D = α /α R k+ R for. The P max D β log 0 as. Proof. Sep. Le α = max{0, α /α} ad ρ = F β log α, where F x s he cumulave dsrbuo fuco of X. By codo.3 ad Proposo A.2 Appedx, here exss cosas r 2 > r > 0 ad ς > 0 such ha r log ς < log ρ < r 2 log ς as s suffcely large. Se { m = m [log ], [ 2 ]} log ρ for all. The m + as. Noe ha max m D m α R k+m log α R k+m. The, by he fac ha LR = LF e S for all. P max D β log P R k+m β log α m { P S k+m log F β log α }, whch s bouded by P S k+m 2m by he defo of m. Ths probably goes o zero by he weak Law of Large Numbers. Hece, o prove he lemma, suffces o show ha P max m D β log

8 as. Sep 2. We ow prove 2.3. Noce P max m D β log m As 2.3 ad 2.5, here exss [S, S +k ] such ha P D β log 2.4 α /α D = β S k+ S He θ 2.5 where Hx s as 2.4. Trvally, x max{ a, b } f a x b for ay a, b R. Thus, f / 2 log /, he eher S / 2 log / or 2 log S k+ S k+ k + + k k + S k+ k + k + + k, whch ur mples ha S k+ /k + logk + / k + as s suffcely large. Thus, og ha Eξ = ad Varξ =, by of Lemma A.2, P 2 log 2 max P S j jk+ j log j e log 2 /3 j as s suffcely large. 2.6 By codo 2., here exss δ 0, ad K 0, + such ha Hx K ωx as 0 < x < δ. Obvously, f ɛ 0, /2 ad x < ɛ, we have ha x 2 x. Thus, f / log /, he log ad / 2log / as s suffcely large, use 2. ad he equaly ha ab a b + a + b o ge α /α He θ K K α ωe θ + + K K ωeθ α log = o O + o + O log log e θ log log e θ log = o log for all m ad s suffcely large, where K α = 2 /α α /α. We use he fac ha ωx loglog x 0 as x + he frs equaly above. By 2.6, α /α P He θ e log 2 /3 log a.s. for all m as s suffcely large. From 2.4, 2.5 ad he above, we have ha P D > β log P S k log + + e log 2 /3 log m 2.7 8

9 for all m as s suffcely large. By L Hospal s rule, oe ca easly check ha P S k > = x k e x k! dx k e k! 2.8 as +. Recallg 2.4, we oba ha P max m D β log 2 log k e log +o + m e log 2 /3 as s suffcely large. The mddle erm above goes o zero evdely; sce m + ad + e log x2 /3 dx <, he las erm s bouded by 3 e log m2 /3 + m e log x 2 /3 dx 0 as +. We fally ge 2.3. Proof of Theorem 3. By Lemma 2., s eough o prove ha P max [ βd log + k loglog + βx exp ] k! e βx 2.9 as. From 2.5, { α /α βd = S k+ S + S k+ S He θ }. By ragle equaly, From Lemma A.4, max [ βd max ] [ S k+ S ] α /α k max ξ max +k He θ. P max [ ] [ ] S k+ S log + k loglog + βx = P max S k+ S log r + k loglog r + βx + o r exp k! e βx as, where r = [ ]+. Secod, s kow ha max ξ / log probably. Thus, o prove he heorem, we oly eed o prove α /α log max He θ [ ] 9

10 probably. Aga, recall ha ab a b + a + b for ay a, b R. As he argume bewee 2.0 ad 2.2, o show 2.20, suffces o demosrae ha log log max [ ] max [ ] He θ 0 probably; probably 2.22 as. By 2.6, P 2 log e log 2 /3 as s suffcely large. Obvously, f x < δ ad δ < /2, he x < 2δ. Therefore, P 4 log P 2 log e log 2 / as s suffcely large. If / 4log / for all [ ], he log max [ 5log 2 ] < ɛ 2.24 /4 for ay ɛ > 0 whe s suffcely large. I follows ha P log max [ ] ɛ max [ P ] 4 log e log 2 /3 as s suffcely large. So 2.22 follows. Now we prove 2.2. By codo 2., here exss K > 0 ad δ 0, such ha Hx K ωx as 0 < x < δ. Thus, f / 4log / for all [ ], he 4 log /2 for all [ ] as s suffcely large. Therefore, log max [ ] K log max [ ] He θ loglog e θ 3K sup { ωx loglog x} 0 x e /2 max [ { ωe θ loglog e θ } ] as goes o + by he gve codo. Fally, by 2.23, P log max [ He θ > ɛ ] P > 4 log [ ] e log[ ] 2 /3 0 0

11 as +. The assero 2.2 s yelded. Proof of Theorem 2. We wll ex show ha lm sup lm f W log β W log β Upper boud. Gve ɛ > 0. Le γ = /2 m{α, }. We clam ha a.s a.s P U + ɛβ log e log γ 2.27 as s suffcely large, where U = max k { α /α D } ad k = [log γ ]. Noe ha max k { α /α D } log τ F e S k+k, where τ = max{α γ/α, 0}. Obvously, τ 2γ/α. The, for e, P max { α /α D } + ɛβ log k P F e S k+k + ɛβ log 2γ/α P S k+k log F + ɛβ log 2γ/α By codo.4 ad Proposo A.2, here exss cosa K > 0 such ha F + ɛβ log 2γ/α exp K log 2γ as s suffcely large. Takg log for boh sdes, we have from of Lemma A.2 ha he las probably s bouded by P S k+k/k +k e 2 2e Ie2 k. I s easy o check ha Ix = x log x for x > 0. Clam 2.27 he follows sce Ie 2 4. Now, le V = max k D, where D = α /α R k+ R. For ay ɛ > 0, P V + 2ɛβ log From 2.3, here exss [S, S +k ] such ha D = β S k+ S He θ =k P βd + 2ɛ log α /α where Hx s as 2.4. Wre α /α βd = S k+ S + S k+ S He θ The, by he equaly ab a b + a + b, we have ha α /α Heθ B C + B + C 2.30 where B = He θ ad C = / α /α. Thus P βd + 2ɛ log P S k + ɛ log + P S k+ S B C + B + C ɛ log 2.3

12 for ay k. We clam ha here exss a cosa K 2 > 0 such ha P B K 2 e log 2 /3 ad P C > K 2 log e log 2 /3 log as suffcely large. Ideed, as 2.23, we have ha P 4 log e log 2 / as s suffcely large. Revewg he argume 2., / α /α K 3 / 4K 3 log / f / 4log / ad s suffcely large, where K 3 = 2 α α. Therefore P C > 4K 3 log e log 2 / as s suffcely large. So he secod equaly of 2.32 follows for ay K 2 4K 3. Now, by 2.6, P / 2log / e log 2 /3 f s large eough. If / 2log / ad s large, by codo 2. ad he assumpo o ωx, here exss a cosa K 4 > 0 such ha B K 4 loglog e θ 2K 4 / log as s suffcely large. I follows ha P B 3K 4 e log 2 /3 log as s large eough. Now akg K 2 = 4K 3 + 3K 4, we ge he frs equaly of Now, f B K 2 / log ad C K 2 log /, he B C + B + C K 5 / log for some cosa K 5 > 0 as s suffcely large. Therefore P B C + B + C > K 5 2e log 2 /3 log 2.36 as s suffcely large. The, for k, og ɛ/k 5 log log k > + ɛ log as s suffcely large, P S k+ S B C + B + C ɛ log P S k + ɛ log + 2e log 2 / as s large eough. From 2.8, P S k > +ɛ log ɛ/2 as s large eough. Combg 2.3 ad 2.37, we ge for all k ad large. By 2.28, 3 P βd + 2ɛ log + 2e log +ɛ/2 P V + 2ɛβ log 3 ɛ/2 + 2 as s suffcely large. The las sum s domaed by e log k2 /3 + e log k2 /3 + =k + e k 2 2 /3 e log 2 /3 =k e log x2 /3 dx log x 2 /3 dx.

13 Use log x 2 log k log x for x k o oba ha log x e 2 log /3 dx x log k/3 k dx = k log k/3, k k 3 whch s bouded by e log k2 /4 as s suffcely large. Combg all he above, keep md ha k = [log γ ], we have P V + 2ɛβ log 5e log k2 /4 e log log 2 /5 as s large eough. Nog W max{u, V }. The las equaly ogeher wh 2.27 cocludes ha P W + 2ɛβ 2e log log 2 /5 log as s suffcely large. Take p = [e ]+ for all. The 2 P W p / log p +2ɛβ 2 2 e log 2 /5 <. By he Borel-Caell Lemma, lm sup W p log p + 2ɛβ as. Observe ha W s o-decreasg. a.s. For ay k 3, here exss such ha p k < p +. The W k / log k W p+ / log p + log p + /log p for all p k < p +. Sce log p + /log p as, we eveually ge for ay ɛ > 0. Ths gves lm sup k Lower boud. By 2.29 ad 2.30, W k + 2ɛβ a.s. log k S k+ S βd + S k+ S B C + B + C for ay. Se γ = max B C + B + C. The max S k+ S 2.38 max βd + max S k+ S γ 2.39 whch gves max βd γ max S k+ S. By 2.36, P γ K 5 log max P B C + B + C K 5 log e log 2 / as s suffcely large. Thus, for ay ɛ 0, /4, P W 2ɛβ log P max βd 2ɛ log P γ max S k+ S 2ɛ log. 3

14 Cosderg he eve he las probably by dsgushg γ K 5 / log or o, ad ocg 2ɛ K 5 / log 2ɛ as. By 2.40, P W 2ɛβ log e log 2 /3 + P max S k+ S ɛ log as s suffcely large. Sce {ξ ; } are..d. radom varables, P max S k+ S ɛ log P max S k+ S ɛ log [ ]+ P max S j+k S jk ɛ log jq where q = [ + /k]. By depedece, he above probably s decal o From 2.8, P S k > ɛ log q e qp S k> ɛ log. P S k > ɛ log ɛk k! log k ɛ 2.4 as. So here exss a cosa K 6 > 0 depedg o k oly such ha q P S k > ɛ log K 6 ɛ for large eough. Collecg all he above, we have P W 2ɛβ log e log 2 /3 + e K6ɛ as s suffcely large. Evdely, he sum of he rgh had sde over all 2 s fe. By he Borel-Caell lemma, lm f W 2ɛβ a.s. log for ay ɛ 0, /4. Leg ɛ 0 +, we fally oba Appedx I hs seco, we frs show ha codo.4 holds for a grea class of probably dsrbuos. The we provde a example ha Theorem does o hold whe codo.4 s volaed. A las we collec some ools ad echcal resuls used Seco 2. PROPOSITION A. Le κ > 0, c > 0, α > 0 ad A [, + be cosas. Le bx, x > A, be a fuco such ha he desy fuco of X s px = ce κ x α +bx I{x > A}. The followg hold. 4

15 If bx = 0 for all x A, he codo.4 holds wh β = ακ /α ad ωx = Olog log x/ log x as x +. If bx s a wce dffereable fuco such ha b x = Ox α / log x ad b x = ox α 2 as x +, he codo.4 holds wh β = ακ /α ad ωx = O/ log log x as x +. If, furher, b x = ox α / log x as x +, he β = ακ /α ad ωx = o/ log log x as x +. The ex resul s a weak coverse of codo.4. PROPOSITION A.2 Suppose codos.3 ad.4 hold. 0,, e rαxα /β F x e sαxα /β as x s suffcely large. The, for ay r > ad s Now we sar o prove he above wo proposos. The followg lemma wll be used for he proof of Proposo A.. LEMMA A. Le κ > 0, c > 0, α > 0 ad A [, + be cosas. Le bx be a wce dffereable fuco such ha b x/x α 0 ad b x/x α 2 0 as x +. Suppose px = ce κ x α +bx I{x > A} s a probably desy fuco. The F x = px + x b x καx α + O x α as x +, where F x = x p d for x > A. A Proof of Lemma A.. Frs, F x = + x p d for x > A. Secod, observe ha p x = px dlog px/dx. So px = p x dlog px/dx. I s rval o verfy ha dlog p d = κα α + b ad d2 log p d 2 = καα α 2 + b 3. for > A. Sce b x/x α 0 as x +, for ay ɛ > 0, here exss B > max{a, 0}, such ha ɛx α b x ɛx α for all x B. Iegrag he hree erms over [B, ], he dvdg α, ad leg +, ad leg ɛ 0 +, we oba ha bx/x α 0 as x +. Ths mples p dlog p/d 0 as + by 3.. By egrao by pars, we have ha d log p F x = p d x d d log px d 2 2 log p d log p = px + p dx d 2 d. d Recall F x = + p d. We have ha x d log px F x + px D F x dx as x > max{a, 0}, where d 2 2 log p d log p D := sup d 2 d = Ox α x x 5

16 as x +, by 3., ad codos b x = ox α ad b x = ox α 2 as x +. Combg he las wo asseros, we oba ha d log px F x = + Ox α px dx as x +. Now dlog px/dx = κα x α + Ob xx α as x +. Thus, by codo b x/x α 0 as x +, we have ha F x = px + x b x καx α + O x α as x +. Proof of Proposo A.. We eed o show ha g log α /α loglog/ ακ = O /α log/ as 0 + for case, ad g log α /α ακ = O /α log log/ as 0 + for case, ad f b x = ox α / log x as x +, he 3.3 sll holds f he O o rgh had sde s replaced by o. Frs, by Lemma A., log F x = κx α bx + Olog x 3.4 as x +. Thus, by Lemma A. aga, for boh case ad case, as x +. = = α /α F x log px F x + x b x bx + log x + O ακ /α x α + O x α log x + bx + x b x ακ /α + O x α α /α Revewg 3.2, le x = F, he = F x. Therefore, 3.2 s equvale o ha F x px α /α log log log F x F x ακ = O /α log F x as x +. By 3.4, he rgh had sde above s Olog x/x α as x +. Therefore he above follows by 3.5 ad he assumpo bx = 0. Idey 3.3 s equvale o ha F x px log α /α F x ακ = O /α 6 log log F x

17 as x +. We frs clam ha codo b x = Ox α / log x mples ha bx = Ox α / log x as x +. If hs s rue, he rgh had sde of 3.6 s equal o O/ log x from 3.4, ad he lef had sde s O/ log x by 3.5. Ths meas ha 3.6 holds. Now we prove our clam. Sce b x = Ox α / log x as x +, here exss x 0 > e ad K > 0 such ha b x Kx α / log x as x x 0. I follows ha bx bx 0 + K x x 0 α log d for x x 0. By L Hospal s rule, s easy o check ha lm x + x x 0 α log d x α log x = α. Therefore, bx = Ox α / log x as x +. Smlarly, codo b x = ox α / log x as x + mples ha bx = ox α / log x as x +. The he secod clam follows. Proof of Proposo A.2. We wll frs prove ha log /α F αβ 3.7 as 0 +. Le ɛ 0,. Noe ha F = /g for 0,. By he gve codo, here exss δ 0, depedg o ɛ such ha ɛ log α/α df + ɛ β d β log α/α for all 0 < δ. Iegrag he above over [, δ], we oba from he secod equaly ha δ F F δ + ɛ log α/α ds βs s for ay 0 < δ. Now log s /α = αs log s α/α. Thus, F F δ + + ɛαβ log /α + ɛαβ log /α. δ Leg 0 + frs, he ɛ 0 +, we oba ha lm sup 0 + F log /α α β. Smlarly, we have lm f 0 +log /α F αβ. Thus, 3.7 follows. Gve η 0, αβ, by 3.7, here exss b 0, such ha αβ η log /α F αβ + η log /α for ay 0, b]. The secod equaly says ha F αβ + η log /α. Le x = αβ + η log /α. The, = exp x α /αβ + η α. Thus F x e xα /αβ +η α for x αβ + η log b /α. Smlarly, we ge F x e x α /αβ η α η log b /α. The Proposo s proved because η 0, αβ s arbrary. for x αβ 7

18 PROPOSITION A.3 Le X sasfy he log-ormal dsrbuo, ha s, log X N0,. The he cocluso of Theorem does o hold. However, / = δ coverges o he Gamma D dsrbuo wh desy px = 2 k x k e 2x Ix 0/k!, where D = /2log R k+ log R for. log X Proof. Sce X = e for, ad {log X ; } are..d. N0,, we have ha e X L, e XL2,, e XL,, LR, R 2,, R = L where X, X 2, are..d. N0, -radom varables. The D = /2log R k+ log R,, has he same dsrbuo as ha of 2 /2 XLk+ 2 /2 XL,. Nocg / 2N0, has desy px = π /2 e x2. By of Proposo A. ad Theorem, α = 2, κ =, ad δ D coverges weakly o G a.s. 3.8 = wh desy px = 2 k x k e 2x Ix 0/k!. If Theorem holds, he here exss β R such ha I β R k+ R x G 2 x a.s. 3.9 = for ay x R as, where G 2 x s he cumulave dsrbuo fuco of a Gamma dsrbuo. By he Mea-value heorem, Therefore, recallg he defo of D, A : = = We clam ha log R k+ log R R k+ R R. = lm f + I β R k+ R x β /2 R I D 2 x. 3.0 R e = + a.s. 3. If hs s rue, by 3.8 ad 3.9, here exss Ω such ha P Ω =, e R ω +, I D ω 2 /2 x G 2 /2 x ad 3.2 = I β R k+ ω R ω x G 2 x 3.3 = as for all ω Ω ad all raoal umber x. Fx eger m, ω Ω, ad raoal umber x > 0. The here exss 0 such ha x β /2 R ω m for all 0. Thus A ω = A I β R k+ ω R ω m G 2 m = 0 8

19 by 3.3 as. Sce m s arbrary, lm A ω =. However, by 3.0 ad 3.2, lm sup A ω G 2 /2 x <. Ths yelds a coradco. Now we prove 3.. Noe ha R ad F e S have he same dsrbuo, where F x = Φlog x for x > 0, ad S = ξ + + ξ,, ad ξ s are..d. Exp-dsrbued radom varables. Recallg ha Φx / 2πxe x2 /2 as x +, we have ha F x 2π log x e log x2 /2 as x +. Therefore, log F e /2 as +. I follows ha P R e = P S log F e S P 2 3 e K 3.4 for large eough, where K > 0 s a cosa resuled usg of Lemma A.2 below he las sep; we also use he fac ha Eξ = ad ha Ix s posve, o-creasg for x 0, ha sep. Thus, R P e <. Ths leads o he desred cocluso by he Borel-Caell lemma. The frs par of ex lemma s c of Remarks o page 27 from [0]; The secod par correspods o Theorem 3.7. o page 09 from [0] whe d = ad C = σ 2. LEMMA A.2 Le {ξ, ξ, =, 2, } be a sequece of..d. radom varables. Le S = = ξ,. The For ay A R ad, P S / A 2e IA, where Ix = sup R {x log Ee ξ } ad IA = f x A Ix. Assume furher ha Eξ = 0, VarX = σ 2 > 0 ad Ee 0ξ < for some 0 > 0. Le {a ; =, 2, } be a sequece of posve umbers such ha a 0 ad a as. The { } lm a a x 2 log P S A = f x A 2σ 2 for ay subse A R such ha f{ x ; x A } = f{ x ; x Ā}. The followg Posso approxmao heorem s a specal case of Theorem from [2], whch s aga a specal case of he Se Posso approxmao mehod, see [3], [24], [25] ad leraures here. Oe applcao of he heorem s sudyg behavors of maxma of radom varables; see, for example, [7], [8] ad [9]. LEMMA A.3 Le I be a dex se ad {B α, α I} be a se of subses of I, ha s, B α I for each α I. Le {η α, α I} be radom varables. For a gve R, se λ = α I P η α >. The P max α I η α e λ λ b + b 2 + b 3 9

20 where b = P η α > P η β >, α I β B α b 2 = P η α >, η β >, α I α β B α b 3 = E P η α > ση β, β / B α P η α >, α I ad ση β, β / B α s he σ-algebra geeraed by {η β, β / B α. Parcularly, f η α s depede of {η β, β / B α } for each α, he b 3 = 0. LEMMA A.4 Le ξ, ξ, ξ 2, be..d. Exp-dsrbued radom varables wh S j = ξ + ξ ξ j, j. Gve eger k, defe W = max{s k+ S, S k+2 S 2,, S k+ S } for. The P W log + k loglog + x exp e x k! for ay x R as +. Proof. Se I = {, 2,, } ad B = {β = {j +, j + 2,, j + k}; β { +, + 2,, + k} } ; η = S k+ S ad = = log + k loglog + x. By L Hospal s rule, oe ca easly check ha as +. The λ = = P S k > = x k e x k! dx k e k! P S k+ S > = P S k > k e k! e x k! as. Recall Lemma A.3, by depedece, b 3 = 0. Noe ha #B 3k. We have from 3.5 ha as. Moreover, b 3kP S k > 2 = O k b 2 3k max 2k+ P S k+ S >, S k+ S > 3kP S k+2 S 2 >, S k+ S > = 3kP S k+ S >, S k > 20

21 where he secod equaly s obaed by Lemma 2. from [8]. Now we esmae he las probably. Frs, by depedece, P S k+ S >, S k > = E P S k > ξ 2,, ξ k 2 = E P S k > ξ 2,, ξ k 2 IA + IA c where A = {S k S > }. If k =, P A = 0. Oherwse, S k S Gammak. I he follows from 3.5 ha he above s decal o By L Hospal s rule, P A + Ee 2 S k+s IS k S k 2 e 0 y k 2 e y dy k 2 e as +. I summary, we have from 3.6 ha b 2 9k k 2 e = O k 2! + e 2 0 y k 2 e y k 2! e2y dy k 2 e + e 2 y k 2 e y dy. = O log as. Cosequely, b 0 as for =, 2, 3. The lemma s cocluded by Lemma A.3. 0 Refereces [] Arold, B.C., Balakrsha, N. ad Nagaraja, H.N Records, Wley, New York. [2] Arraa, R. ad Goldse, L. ad Gordo, L Two Momes Suffce for Posso Approxmao: The Che-Se Mehod. A. Probab. 7, [3] Barbour, A.D., Hols, L. ad Jaso 992. Posso Approxmaos. Oxford Sudes Probably 2, Claredo, Oxford, 992. [4] Bose, A., Gagopadhyay, S., Maulk, K., ad Sarkar, A Covergece of al sum for records. Exremes. 9, [5] Bose, A., Gagopadhyay, S. ad Sarkar, A Paral sum process for records. Exremes. 8, [6] Chadler, K. N The dsrbuo ad frequecy of record values, Joural of he Royal Sascal Socey, Seres B, 4, [7] Davd, H. A. ad Nagaraja, H. N Order Sascs, 3 edo, Wley-Ierscece. [8] Deheuvels, P. 98. The srog approxmao of exremal processes, Z. W. Verw. Gebee, 58, -6. 2

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