The point process approach for fractionally differentiated random walks under heavy traffic

Transcription

1 Avalable ole at Stochastc Processes ad ther Applcatos 122 (212) The pot process approach for fractoally dfferetated radom walks uder heavy traffc Ph. Barbe a, W.P. McCormck b, a CNRS (UMR 888), 9 rue de Vaugrard, 756 PARIS, Frace b Departmet of Statstcs, Uversty of Georga, Athes, GA 362, USA Receved 2 March 211; receved revsed form 17 August 212; accepted 22 August 212 Avalable ole 31 August 212 Abstract We prove some heavy-traffc lmt theorems for some ostatoary lear processes whch ecompass the fractoally dfferetated radom walk as well as some FARIMA processes, whe the ovatos are the doma of attracto of a o-gaussa stable dstrbuto. The results are based o a exteso of the pot process methodology to lear processes wth osummable coeffcets ad make use of a ew maxmal type equalty. c 212 Elsever B.V. All rghts reserved. MSC: prmary 6F99; secodary 6G52; 6G22; 6K25; 62M1; 6G7; 62P2 Keywords: Heavy traffc; Pot process; Supremum fuctoal; Fractoal radom walk; FARIMA process; Posso process 1. Itroducto ad ma result The purpose of ths paper s to study heavy traffc approxmatos for some lear ostatoary processes havg log rage depedece ad ovatos the doma of attracto of a o-gaussa stable dstrbuto. The motvato for ths study stems from a terest such topcs as the global maxmum of stochastc processes, boudary crossg ad related problems. Whle Barbe ad McCormck [2] deal wth lear processes for whch the Correspodg author. Tel.: E-mal addresses: phlppe.barbe@math.crs.fr (Ph. Barbe), wpmcc@uga.edu, bll@stat.uga.edu (W.P. McCormck) /$ - see frot matter c 212 Elsever B.V. All rghts reserved. do:1.116/j.spa

2 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) ovatos are magfed over tme, the curret paper deals wth dampeed ovato, but at a rate of dampeg slow eough to be o-summable. Whereas covergece to fractoal Lévy process was a key tool used [2], here the relevat covergece result s that of covergece to the pot process. Though the pot process approach to heavy traffc approxmato s well uderstood the case of the radom walk (see e.g. [1, Chapter 8]) ad for sequeces wth rather short rage depedece, the exstg methodology fals the settg of ths paper, because of the slow decay of coeffcets of the lear processes uder cosderato. To descrbe the settg, we cosder a power seres, g(x) = g x, whose radus of covergece s 1. Gve a dstrbuto fucto F, we ca defe a (g, F)-process (S ) as follows. Let (X ) 1 be a sequece of radom varables, depedet, havg commo dstrbuto fucto F. We set X = f s opostve. We defe the backward shft operator B by B X = X 1 ad set S = g(b)x = g X. < Importat examples of such processes clude radom walks, autoregressve movg averages processes ad ther fractoal tegrated extesos. I partcular, f g(x) = (1 x) γ, the S = (1 B) 1 γ (1 B) 1 X s the radom walk (1 B) 1 X dfferetated 1 γ tmes. I order to develop a pleasg theory whch s applcable to some ostatoary FARIMA processes, we assume that the sequece (g ) s ultmately postve ad regularly varyg of egatve dex γ 1 wth γ (, 1). Ths forces the fucto g to dverge to + as ts argumet teds to 1. Our earler work, Barbe ad McCormck [2], cocetrates o the case where γ s greater tha 1, forcg (g ) to dverge to fty. I cotrast, ths curret paper, we assume that γ s less tha 1, forcg (g ) to coverge to. For ay postve t defe the partal sum g [,t) = g. <t Uder (1.1), the sequece (g [,) ) 1 dverges to fty. A heavy traffc approxmato descrbes the lmtg behavor of some fuctoal of the process whe the expectato of the ovatos teds to. Wrtg S = (S g [,) EX 1 ) + EX 1 g [,), a alteratve vewpot s to cosder that (1.1) F s cetered (1.2) ad seek the lmtg behavor of a fuctoal of the process (S ag [,) ) whe a teds to. As dcated [13] ths problem has bearg o heavy traffc lmts queueg theory, as well as some movg boudary crossg probablty problems. Throughout the paper we wll use c for a geerc costat whose value may chage from place to place.

3 43 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) We are terested the heavy-tal stuato where F belogs to the doma of attracto of a stable dstrbuto wth dex α (1, 2). (1.3) Assumpto (1.3) mples that F s tal balaced the followg sese. Wrtg F for the dstrbuto of X 1 ad M 1 F for that of X 1, there exst p ad q both [, 1], such that F pf ad M 1 F q F at fty; ths last equalty beg read as M 1 F = o(f ) f q vashes. These asymptotc relatos mply p + q = 1. Uder (1.3), F s regularly varyg of dex α; ad so s F, M 1 F, f p, q, does ot vash respectvely. For smplcty, we wll assume throughout ths paper that p does ot vash. If q vashes we wll also assume that the lower tal of the dstrbuto fucto F decays slghtly faster tha the upper oe the sese that for some costat c M 1 F(t) cf(t log t) log t ultmately. (1.4) Assumpto (1.3) gves rse to a Lévy measure ν whose desty wth respect to the Lebesgue measure λ s dν dλ (x) = pαx α 1 1 (, ) (x) + qα( x) α 1 1 (,) (x). The fucto k(t) = F g [,t) (1 1/t) wll play a role our results ote that the meag of the otato k ths paper s dfferet tha that [2], but wll play at tmes, but ot always, a aalogous role. Gve (1.1) ad Karamata s Taubera theorem for power seres [3, Corollary 1.7.3] g(1 1/t) k(t) Γ (1 + γ )F (1 1/t) as t teds to fty. I partcular, k s regularly varyg at fty, wth dex γ 1/α. It teds to fty at fty whe αγ > 1 ad to whe αγ < 1. It s asymptotcally equvalet to a mootoe fucto whe αγ 1. Our frst result s a pot process form of a heavy-traffc approxmato. It ecompasses the sprt of ths paper, though by o meas ts techcal aspects. I order to state t, we eed to specfy the space whch we cosder pot processes. For ths purpose, gve a topologcal space A, we wrte M(A) the space of all Rado measures o A, edowed wth the topology of vague covergece. We wrte R for R \ {}. Fally, throughout ths paper, Π = δ (t,x ) M [, ) R 1 s a Posso pot process wth mea testy λ ν. Theorem 1.1. Assume that (1.1) holds for some postve γ less tha 1, that (1.2) ad (1.3) hold, ad that k teds to fty at fty. The dstrbuto of the pot process δ k (1/a), S ag [,) as elemet M (, ) R coverges to that of 1 j δ t, g j x t γ as a teds to.

4 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) The ma lmtato of ths theorem s that M (, ) R s edowed wth the topology of vague covergece. As a cosequece, t s ot possble to derve by a smple applcato of the cotuous mappg theorem a heavy traffc approxmato o the overall supremum of the process or the tme spet over the boudary ag [,), because those cocer global features of the process. To go from a local theorem such as Theorem 1.1 to some global result, the stadard tool the statoary settg s a form of Kolmogorov s maxmal equalty. I the curret cotext, o such equalty s avalable. The followg result s the ma techcal ovato of ths paper ad seems useful beyod the corollares that we wll state. It asserts that wth hgh probablty S caot exceed ag [,) after a tme of order k (1/a) whe a s close to ad αγ > 1. Theorem 1.2. Assume that (1.1) (1.3) hold, ad that αγ > 1. If q vashes, assume furthermore that (1.4) holds. The lm T lm sup P > T k (1/a) : S > ag [,) =. a Theorems 1.1 ad 1.2 allow us to obta some heavy traffc approxmatos whch parallel some tal approxmatos obtaed by Braverma et al. [4] the cotext of Lévy processes. The frst asserto of our frst corollary asserts that for a heavy traffc approxmato to the overall supremum of the process to make sese we should have αγ 1. It also mples that the cocluso of Theorem 1.2 caot hold f αγ < 1, or, more geerally, f lm sup t k(t) <. Corollary 1.3. Assume that (1.1) (1.3) hold. If q vashes, assume furthermore that (1.4) holds. () If lm sup t k(t) <, partcular f αγ < 1, the for ay postve a, sup(s ag [,) ) = + almost surely. () If αγ > 1, the dstrbuto of sup(s ag [,) ) F 1 k 1 (1/a) coverges to that of sup 1 (g j x t γ ). The latter radom varable s almost surely fte. j Note that Corollary 1.3 leaves ope the boudary case where αγ = 1 ad lm t k(t) = +. It s cocevable that the cocluso of () remas, but we do ot kow how to prove t. Corollary 1.3 also leaves ope the seemgly less terestg stuato where k oscllates such a way that lm f t k(t) = ad lm sup t k(t) = +. Our ext result cocers the last crossg of the boudary ag [,), τ a = sup{ : S > ag [,) }, wth sup = say. It asserts that properly ormalzed, ts dstrbuto coverges to a Fréchet oe. We wll use the otato g = max g ad g = f g. Note that the defto of g the fmum s a mmum f ad oly f at least oe g s opostve ad that g s always opostve uder (1.1).

5 432 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Corollary 1.4. Uder the assumptos of Corollary 1.3(), for ay oegatve y, lm P τ a a k (1/a) y = exp pgα + q g α 1. αγ 1 yαγ 1 The ext two corollares show that our techque fals to delver a useful result o the tme spet over the boudary, 1{S > ag [,) } as a teds to, but succeeds quatfyg how far the process s from the boudary average. I order to clarfy that asserto ad state some formal results, we wrte G for the coutg measure assocated to the sequece (g ), that s, G = δ g. Corollary 1.5. Uder the assumptos of Corollary 1.3(), for ay postve θ, the dstrbuto of 1{S > ag [,) } θk (1/a) coverges to that of t >θ G[tγ /x, ) as a teds to. The lmtg dstrbuto s odegeerate. However, t degeerates to a pot mass at fty whe θ teds to because for ay postve u < v, there are almost surely ftely may pots (t, x ) such that t γ /x belogs to (u, v). I partcular ths shows that lm 1{S > ag [,) } = + a 1 probablty, ad eve almost surely sce 1{S > ag [,) } s mootoe a for ay larger tha some determstc uder (1.1). However, gve Corollary 1.5, 1{S > ag [,) } = o k (1/a) (1.5) 1 probablty as a teds to ; deed, for ay postve θ, 1{S > ag [,) } θk (1/a) θk (1/a) ad Corollary 1.5 mples that 1{S > ag [,) } = O(1) = o k (1/a) θk (1/a) probablty as a teds to ; sce θ s arbtrary, ths mples (1.5). Thus, order to study the overall tme spet over the boudary, 1{S > ag [,) }, Corollary 1.5 forms us that t s eough to look at the process S over a tme of order o(k (1/a)) ad that we have some dea o a upper boud o how fast ths tme grows as a teds to. Whle Corollary 1.5 gves oly a partal formato o the tme spet over the boudary (ag [,) ), our ext result forms us o how far the process s from ths boudary some average measure, parallelg a smlar result of [4] the cotext of Lévy processes.

6 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Corollary 1.6. Let ρ be a postve real umber. Uder the assumptos of Corollary 1.3(), the dstrbuto of 1 ρ (S ag [,) ) ρ F 1 k 1 + (1/a) coverges to that of, j (g j x t γ ) ρ + as a teds to. Gve Corollary 1.6, the questo arses as to whe the lmtg dstrbuto s odegeerate, ad ths s settled the ext result. For ths purpose, gve a postve ρ, we defe the fucto h ρ (t) = g g ρ. t + Ths fucto s cotuous, creasg o [1, ) ad vashes o [, 1]; uder (1.1), t s bouded f ad oly f ( g ρ ) s summable. Furthermore, let ρ = αγ 1 αγ (1 γ ). Proposto 1.7. The radom varable, j (g j x t γ ) ρ + s fte wth probablty or 1. It s almost surely fte f ad oly f αγ > 1 ad ether () ρ > ρ, or () ρ = ρ ad h ρ (t) = o(t 1/αγ (γ 1) ) as t teds to fty ad h ρ (y) α(1 γ ) y 1+1/γ dy <. Note that f ρ s less tha ρ, Proposto 1.7 mples that the lmt volved Corollary 1.6 s fte, meag that the ormalzato 1/F 1 1/k (1/a) ρ s too large. If γ < 1/2, the codto αγ > 1 caot be fulflled sce α s less tha 2. I those cases, as all those left ope by Proposto 1.7, we do ot kow what the proper ormalzato s. 2. Proof of Theorem 1.1 For ay postve teger, set F (1 1/) c, = g [,) x df(x). Sce F s cetered, F (1 1/) F (1/) F (1/) x df(x) = O F (1 1/)/ as teds to fty. Thus, sce γ s less tha 1, for ay postve M, lm max c,/f 1M Cosder the radom measure N = δ (/,S /F 1 (1 1/) =. (2.1) (1 1/))

7 434 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) the space of all measures o (, ) R edowed wth the topology of vague covergece. Theorem 5.2 [1] asserts that the dstrbuto of the radom measure δ (/,(S c, )/F (1 1/)) 1 j δ (t,g j x ). Sce (2.1) holds, ths mples that the coverges weakly to that of N = 1 dstrbuto of N coverges to that of N as well. Theorem 1.1 follows from the covergece dstrbuto of the radom measure N k (1/a) to N, the defto of k, ad the cotuty of the map µ δ (u,x/u) dµ(u, x) o M (, ) R. 3. Proof of Theorem 1.2 Defe Λ = Λ(1/a) by Λ(1/a) = k (1/a). Usg the same argumets gve just pror to (3.6) [2], t suffces to prove that lm T lm P > Λ : S > T g [,) /k(λ) =. (3.1) Λ The proof of (3.1) s structured mostly three steps, ad a fourth oe to hadle the part of the proof dealg wth the lower tal of the dstrbuto. Step 1. Let (a ) ad (b ) be two sequeces dvergg respectvely to ad +. We assume that We set lm b /( a ) s postve or fte. σ 2 = Var X 1 1 [a,b ](X 1 ) ad, for ay postve teger at most, Z, = X 1 [a,b ](X ) EX 1 [a,b ](X ) σ. Up to ceterg, the part of S made by the mddle ovatos s M = σ g Z,. < As [2], we costruct (b ) as follows. Let ( m ) be a regularly varyg sequece of dex β. Set b = F (1 m /). We the defe m = F( b ) ad set b = F (1 m /). Ths costructo esures that the sequece (m ) s regularly varyg wth dex β ad 1 m / s the rage of F, so that the equalty F (1 u) > b s equvalet to u < m /. The sequece (a ) s costructed a aalogous way, swtchg the tals.

8 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Proposto 3.1. For ay postve β less tha 1 ad ay postve T, lm P Λ : M > T g [,) /k(λ) =. Λ Proof. As the proof of Proposto [2], σ cf (1 m /) m / as teds to fty. Moreover, for ay eve postve teger r such that g r coverges, there exsts a postve costat c r such that r E M c r σ. Markov s equalty yelds P M > T g [,) /k(λ) r σ k(λ) cr T g [,) c F (1 m /) m T r k(λ)r g [,) r 1. Ths asymptotc equvalet s of the form k(λ) r tmes a fucto of whch s regularly varyg of dex 1 β r + β 1 1 α 2 γ 1 = r α γ + β α Ths dex s less tha 1. Therefore, by Boferro s equalty, a comparso betwee sum ad tegral, ad Karamata s theorem, P > Λ : M > T g [,) /k(λ) c F (1 m Λ /Λ) m r Λ T r k(λ)r. g [,Λ) Ths boud s regularly varyg Λ of egatve dex rβ (1/2) (1/α) ad teds to as Λ teds to fty. Step 2. We cosder the part of S made by the extreme ovatos, = g X 1 (b, )(X ). T + < The purpose of ths step s to show that our problem we ca gore the cotrbuto of T + the rage of exceedg Λ 1+ϵ. The followg lemma wll be strumetal; t s stroger tha what we eed ths step, but ths stregth wll tur out to be useful the ext step. Lemma 3.2. If β < (1 γ )/(1 1/α) the wheever T s large eough, s at least Λ ad Λ s large eough, ET + T g [,)/k(λ). Proof. The proof of Lemma [2] mples ET + g α [,) α 1 m1 1/α F (1 1/).

9 436 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Substtutg T by a multple of t, t suffces to prove that for ay large eough g [,) m1 1/α < T k() k(λ). Sce k s regularly varyg of postve dex, t suffces to show that for ay large eough g [,) m1 1/α < T. Ths holds because the left had sde s regularly varyg of dex γ 1 + β(1 1/α) whch s egatve. The ma result of ths step 2 s the followg. Proposto 3.3. Let ϵ be a postve real umber. If β < 1 γ 1 ϵ 2γ α 1 + ϵ, the lm T lm sup P > Λ 1+ϵ : T + ET + > T g [,)/k(λ) =. Λ Proof. Let (U ) 1 be a sequece of depedet radom varables uform over [, 1]. Let U be the emprcal dstrbuto fucto of (U ) 1. We wrte (U, ) for the order statstcs of (U ) 1. Wthout ay loss of geeralty, we assume that X = F (1 U ), so that = g F (1 U )1{U < m /}. T + < Sce the uform dstrbuto s cotuous, we have wth probablty 1 that for every, = g F (1 U )1{U m /}. (3.2) T + < Thus, for ay large eough, T + cf (1 U 1, )g [,U (m /)). We wrte Id for the detty fucto ether o [, 1] or (, ) accordg to the cotext. Let (ξ ) 1 be a slowly varyg odecreasg sequece such that 1 1/(ξ ) coverges. From Kefer s [9] Theorem 1 we deduce that U 1, 1/(ξ ) almost surely for large eough, whle Shorack ad Weller s [12] Theorem 2 mples U ξ Id almost surely for large eough. The, usg Potter s boud for F (1 1/Id), usg that (g [,) ) s regularly varyg ad that γ > 1/α, we obta T + cf (1 1/)ξ 2γ g [,m ). Therefore, the equalty T + k(λ) > ct k() g [,m )ξ 2γ. > T g [,)/k(λ) mples, almost surely for large eough, (3.3)

10 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) I ths equalty, the rght had sde s regularly varyg of dex γ (1/α) βγ, whch s postve provded β < γ (1/α) /γ. I ths rage of β ad the rage of at least Λ 1+ϵ, the rght had sde of (3.3) s at least a costat tmes ts value at Λ 1+ϵ. Ths lower boud, as a fucto of Λ, s regularly varyg ad for (3.3) to hold we must have, comparg the dex of regular varato, γ 1 α (1 + ϵ) γ 1 α βγ. Ths does ot hold uder the assumpto of the lemma, ad therefore (3.3) does ot occur. So T + T g [,)/k(λ) almost surely the rage Λ 1+ϵ ad for Λ large eough. Lemma 3.2 mples that ET + T g [,)/k(λ), wheever exceeds Λ 1+ϵ ad Λ s large eough. Ths proves the proposto. Step 3. Gve Lemma 3.2, our goal s ow to show that (Λ, Λ 1+ϵ ) : T + > T g [,)/k(λ) =. (3.4) lm T lm sup P Λ For ths, we approxmate T + by a smpler quatty. It s coveet to wrte N for Λ 1+ϵ. Let (U ) 1 be a sequece of depedet radom varables uformly dstrbuted o [, 1]. Let τ be the radom permutato of {1, 2,..., N} such that U τ() = U,N. Wthout ay loss of geeralty we ca assume that X = F (1 U ). For (Λ, Λ 1+ϵ ), we the have, almost surely as (3.2), T + = g X 1{X > b } 1 = 1N g τ() F (1 U,N )1{U,N m /}1{τ() }. Let (V ) 1 be a sequece of depedet radom varables uformly dstrbuted o [, 1], depedet of (U ). Let G N be the emprcal dstrbuto fucto of (V ) 1N. Wthout ay loss of geeralty we ca assume that τ() = N G N (V ). The, settg we have R 1,,N = T + = : U,N m ; G N (V ) N R 1,,N g N G N (V )F (1 U,N )., Let =,N be R 1,,N such that N G N (V ) s mmum whe s. Such exsts ad s well defed because R 1,,N s fte ad for R 1,,N the dffereces N G N (V ) are oegatve ad assume, almost surely, dfferet values for dfferet. Set T + 1,,N = g N G N (V )F (1 U,N ). Our ext lemma shows that we ca approxmate T + by T + 1,,N order to prove (3.4).

11 438 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Lemma 3.4. For ay ϵ ad β small eough, for ay postve T, lm P (Λ, Λ 1+ϵ ) : T + T + 1,,N > T g [,)/k(λ) =. Λ Proof. Wrtg R 1,,N for the cardalty of R 1,,N, we have T + T + 1,,N = g N G N (V )F (1 U,N ) R 1,,N \{ } R 1,,N max R 1,,N \{ } g N G N (V ) max F (1 U,N ). (3.5) R 1,,N Let η be a postve real umber less tha 1. As [2], let (W ) be a radom walk whose cremets are stadard expoetal radom varables, ad wrte U,N as W /W N+1. Lemma [2] shows that max R 1,,N = O P (m N log N) ΛN as Λ teds to fty. Robbs [11] proved that provded c s small eough, the set Ω = {U,N c/n : 1 N} has probablty at least 1 η. A teger R 1,,N s such that U,N m /, ad o Ω we obta cm N/. So, f s R 1,,N \ { } ad Ω occurs, V V m 2 jcm N/ V j, cm N/ V j 1, cm N/. (3.6) Devroye s [5] Theorem 3.1 mples that the rght had sde of (3.6) s almost surely at least 2 /cm 2 N 2 log N wheever Λ s large eough. For at least Λ, usg that /m s asymptotcally equvalet to a odecreasg fucto ad hece at least Λ/m Λ, the rght had sde of (3.6) s at least cλ 2(β+ϵ) / log Λ, ad, f β ad ϵ are small eough, domates 1/ N asymptotcally. Sce G N = Id + O P 1/ N by Dosker s [6] varace prcple, m N G c R 1,,N \{ N (V ) G N (V ) 2 } m 2 N log N for all (Λ, N), wth probablty at least 1 η. Thus, wrtg N G N (V ) as N G N (V )+ N G N (V ) G N (V ), t follows that max g R 1,,N \{ N G N (V ) cg } 2 /(m 2 N log N) cg Λ 2 /m 2 Λ N log N. Sce P U 1,N c/n 1 η f c s small eough ad N s large eough, F (1 U 1,N ) cf (1 1/N) wth probablty at least 1 η. Usg Potter s boud, ths s at most cf (1 1/)(N/) (1/α)+η wheever N s large eough.

12 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Thus, wth probablty at least 1 η, (3.5) s at most cm N (log N)g Λ 2 m 2 Λ N log N N (1/α)+η F (1 1/). For ths boud to exceed T g [,) /k(λ) we must have, as Λ teds to fty, cm N (log N)g Λ 2 m 2 Λ N log N N (1/α)+η T k() T. (3.7) Λ k(λ) The left had sde s a regularly varyg of Λ, of dex 1 β(1 + ϵ) + (1 2β ϵ)(γ 1) + ϵ α + η = γ 1 + O(β) + O(ϵ). Ths dex s egatve f β ad ϵ are small eough, ad (3.7) caot hold. Ths proves the lemma. Note that by costructo T + 1,,N s a approxmato of the sum < g X 1{X > b } by a sgle oe of ts summads. Sce each summad s at most max g X,, we see that order to show that lm T lm sup P Λ t suffces to prove that lm T lm sup P Λ (Λ, Λ 1+ϵ ) : T + 1,,N > T g [,)/k(λ) =, (3.8) (Λ, Λ 1+ϵ ) : X, > T g [,) /k(λ) Wrtg X, = F (1 U 1, ), ths amouts to provg that lm T =. lm sup P (Λ, Λ 1+ϵ ) : U 1, F T k() Λ k(λ) F (1 1/) =. (3.9) Let (V ) be a ew sequece of depedet radom varables havg a uform dstrbuto over [, 1]. Wrte (V, ) 1 for the order statstcs of (V ) 1. Settg V 1, = 1, we have d (U 1, ) Λ =(U1,Λ V 1, Λ ) Λ. Applyg Boferro s equalty, we see that for (3.9) to hold t suffces to have lm lm sup P (Λ, Λ 1+ϵ ) : U 1,Λ F T k() T Λ k(λ) F (1 1/) = (3.1) ad, replacg by Λ +, ad settg k(λ + ) v = F T F 1 1 k(λ) Λ + to also have lm T lm sup P (, Λ 1+ϵ ) : V 1, v =. (3.11) Λ

13 44 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) The rght had sde of the equalty volved (3.1) s equvalet to a fucto decreasg. So the rage of betwee Λ ad Λ 1+ϵ t s at most a costat tmes T α F F 1 1 T α Λ Λ. Sce the dstrbuto of ΛU 1,Λ coverges to a stadard expoetal oe, (3.1) holds. To prove that (3.11) holds, we use a blockg argumet. Cosder a real umber θ greater tha 1/(αγ 1) ad for ay teger set = Λ θ. Potter s boud mples as Λ teds to fty ad uformly postve v T α F (1 + θ ) γ (1/α) η F 1 1 Λ(1 + θ ) T α F (1 + θ ) γ 2η F 1 1 Λ T α (1 + θ ) α(γ 3η) 1 Λ. Therefore, for ay Λ large eough ad ay postve, P V 1, v P V 1, ct α (1 + ) θα(γ 3η) 1 Λ 1 1 ct α (1 + ) θα(γ 3η) 1 Λ θ. (3.12) Λ Note that (1 + ) θα(γ 3η) /Λ teds to as Λ teds to fty, uformly oegatve. So (3.12) s at most cλ θ T α (1 + ) θα(γ 3η) 1 Λ ct α θ(1 αγ +3αη). Gve our choce of θ, we see that f η s small eough, the expoet of s less tha 1. Thus, Boferro s equalty mples lm T lm sup P 1 : V 1, v =. Λ If s betwee 1 ad, the V 1, V 1, ad sce v /v 1 s bouded away from ad fty uformly as Λ teds to fty, we proved (3.11) ad (3.9) as well as (3.8). Combg (3.8) ad Lemma 3.4 gves (3.4). Step 4. Let T = < g X 1 (,a )(X ). Note that S T E(S T ) = M + T + ET +. Combg Proposto 3.1, (3.4), ad Lemma 3.2 to hadle T + ET + Λ Λ 1+ϵ, Proposto 3.3 to hadle T + ET + the rage > Λ1+ϵ, we see that lm T lm sup P > Λ : S T Λ Hece, order to prove (3.1), t suffces to show that lm T lm sup P > Λ : T Λ E(S T ) > T g [,)/k(λ) =. ET > T g [,)/k(λ) =.

14 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Ths follows by the very same argumets as Secto 3.5 of [2], substtutg M 1 F for F steps 2 ad 3 f q does ot vash, ad usg a couplg argumet whe q vashes. 4. Proof of the corollares I ths secto we prove Corollares As for provg Theorem 1.1, we cosder Λ depedg o a through the relato k(λ) 1/a Proof of Corollary 1.3 Before provg both assertos, we eed to make some prelmary remarks. Let T be a postve real umber. We obta F F g [,) g [,Λ) S ag [,) S ag [,Λ) (1 1/Λ) = F (1 1/Λ) S = (1 1/Λ) g [,) 1 + o(1) g [,Λ) where the o(1) term s uform betwee ad ΛT ad as a teds to. Let ϵ be a postve real umber. Let N ad N be the pot processes defed the proof of Theorem 1.1. By the Skorokhod Dudley Wchura represetato theorem, we ca costruct a verso of N ad, for each, a verso of N such that ths verso of N coverges almost surely to N as pot measures o [, T ] R. We cosder these versos eve though we use the same otato as the orgal processes. For these versos, S 1{S /F sup (1 1/Λ) > ϵ} ag [,) F (1 1/Λ) 1{ ΛT } (4.1.1) coverges almost surely to sup 1 sup(g j x 1{g j x > ϵ} t γ )1{ t T } j as a teds to. Sce ϵ s arbtrary ad sup (S ag [,) )/F ΛT s wth ϵ of (4.1.1), ths mples lm sup a ΛT F (1 1/Λ) S ag [,) = sup (1 1/Λ) 1 sup(g j x t γ )1{ t T }. (4.1.2) j Wth these prelmares, we ca prove both assertos of Corollary 1.3. Note that the evet {max S ag [,) = + } s the tal σ -feld assocated wth the sequece (X ) 1 ; hece Kolmogorov s 1 law esures that t has probablty or 1. Proof of asserto(). If x s a real umber, we wrte x + for x ad g for max g. For ay oegatve teger r, defe M r = g max (x ) +. :rt <r+1

15 442 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Note that max :rt <r+1 sup j (g j x ) + M r. Let r ad T be some postve tegers wth r < T. We have max (S ) + /g [,) = max <T r<t max r<(r+1) (S ) + F (1 1/) 1 F (1 1/) F (1 1/) k(). Sce our verso of N coverges almost surely to N, ad k( ) s ultmately bouded, we obta max (S ) + /g [,) c max Mr + o(1) 1 <T r r<t 2(r + 1) 1/α almost surely as teds to fty provded r s chose large eough. It the suffces to show that lm max M r /r 1/α = + T 1rT probablty. Sce γ s less tha 1, the sequece (g j ) j has a largest term, g, whch s postve uder (1.1). Let (ω ) 1 be a sequece of depedet radom varables havg a expoetal dstrbuto wth mea 1. The dscusso followg Lemma 6.1 [1], or the calculato betwee (4.1.3) ad (4.1.4) ths paper, show that (M r ) r1 has the same dstrbuto as (p 1/α g ωr 1/α ) r1. Thus, t suffces to show that f r1 rω r = probablty. Ths follows from the equalty P m rω r > ϵ = e ϵ/r 1rT 1rT ad the dvergece of the seres r1 1/r. Proof of asserto (). Gve our prelmary remarks, ad partcular (4.1.2), t follows from Theorem 1.2. It remas to show that the lmtg radom varable volved Corollary 1.3() s almost surely fte. We wrte ν ad ν + for the restrcto of ν to (, ) ad (, ) respectvely. Let Π ad Π + be two depedet Posso processes wth respectve mea measures λ ν ad λ ν +. For a pot process Π = 1 δ (t,x ) wrte Π g for sup 1 j g j x t γ. Sce Π ad Π + are depedet, Π + Π + s a Posso process wth mea testy λ ν. Sce (Π + Π + ) g = Π g + Π g, t suffces to show that Π g + s fte. Wrte Π + = 1 δ (t,x,+ ). Sce (g j ) s bouded, Π g + sup(cx,+ t γ ) 1 (4.1.3) wheever c s at least g. So t suffces to show that the upper boud (4.1.3) s almost surely fte. Sce Π + s a Posso process, the radom varables M r = sup cx,+ r γ, r N, :t [r,r+1) are depedet. Moreover, sup 1 (cx,+ t γ ) sup M r. r

16 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Recall that Π + has testy λ ν +. Sce x + r γ P {M r > x} = P : (t, x,+ ) [r, r + 1), c x + r γ = P Π + [r, r + 1), 1 c x + r γ α = 1 exp p, c we have P sup M r > x r1 1 exp r p c x + r γ α. (4.1.4) Ths seres s coverget sce ts r-th term s equvalet to c/r αγ as r teds to fty ad αγ s greater tha 1. Boudg c/(x + r γ ) by c/(1 + r γ ) whe x exceeds 1, the domated covergece theorem mples that (4.1.4) teds to as x teds to fty, cocludg the proof of Corollary Proof of Corollary 1.4 Let y be a postve real umber ad let T be a real umber greater tha y. Note the equalty of evets {τ a > k (1/a)y} = { > Λy : S > ag [,) } S = : Λy < ΛT ; > 1 ag [,) > { > ΛT : S > ag [,) }. The, combg Theorems 1.1 ad 1.2, we obta that lm P {τ a > k (1/a)y} = P 1{t > y; g j x /t γ a > 1} >, j = 1 P 1{t > y, g j x /t γ > 1} =., j We splt the summato over, j to two, by dstgushg the j for whch g j s postve ad those for whch g j s egatve. The frst sum, whch g j x /t γ exceeds 1, volves the x whch are postve, whle the secod oe those that are egatve. Those two sums are depedet sce the (t, x ) form a Posso process. Furthermore, 1{t > y, g j x > t γ } =, j:g j > f ad oly f 1{t > y, g x > t γ } =,

17 444 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) ad smlarly for the summato over g j egatve. We the obta P 1{t > y, g j x > t γ } =, j = P Π {(t, x) : t > y, g x > t γ } {t > y, g x > t γ } =. Sce Π s a Posso process, ths last probablty s exp EΠ {(t, x) : t > y, g x > t γ } {t > y, g x > t γ }. We the have, EΠ {(t, x) : t > y, g x > t γ } = 1{t > y, g x > t γ }pα dx dt xα+1 = pgα αγ 1 y1 αγ, ad a smlar equalty whe we substtute g for g. Ths proves Corollary Proof of Corollary 1.5 We rewrte the sum volved the corollary as S 1 k θ; > 1. (1/a) ag [,) Ths sum s a tegral wth respect to the pot process volved Theorem 1.1 ad we apply the cotuous mappg theorem wth the help of Theorem 1.2. The, ote that 1{t θ : g j x > t γ }, j we ca sum over j frst, obtag the expresso gve Corollary Proof of Corollary 1.6 Let T deote a postve real umber. I the sum (S ag [,) ) ρ +, we cosder frst the exceedg ΛT. Theorem 1.2 mples that lm T lm sup P a ΛT (S ag [,) ) ρ + = = 1. Sce ag [,Λ) /F (1 1/Λ) 1 as a teds to, F 1 1 p (S ag [,) ) ρ + Λ <ΛT = x g [,Λu) ρ 1 + o(1) 1 [,T ) (u) dn Λ (u, x). g [,Λ) +

18 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Sce N coverges dstrbuto to N wth respect to the vague topology ad g [,) s regularly varyg of postve dex, Potter s bouds ad the domated covergece theorem mply that the above tegral coverges to (x u γ ) ρ +1 [,T ] (u) dn(u, x). After takg lmt T, ths proves the covergece Corollary Proof of Proposto 1.7 Throughout ths subsecto, we set S = (g j x t γ ) ρ +., j The proof of Proposto 1.7 s rather volved ad requres several lemmas. Our frst oe settles the smple stuato where αγ 1. Lemma If αγ 1, the S = + almost surely. Proof. Let Y = { : g x t γ > 1}. Clearly, S Y. The radom varable Y has a Posso dstrbuto wth mea p 1 x > 1 + tγ α dx α g dt = p xα t γ dt. g The tegral dverges f ad oly f αγ 1, whch s equvalet to Y = + almost surely. Gve Lemma 4.5.1, we cocetrate ow o the case where αγ > 1. We wll eed some prelmary results. The followg oe wll be strumetal, ad t s ot sharp o purpose. Lemma Assume that αγ > 1. If g j ρ s fte, the ρ > (αγ 1)/γ. Proof. (g ρ j ) s a regularly varyg sequece of dex ρ(γ 1). For g j ρ to be fte, we must have ρ(γ 1) 1, that s ρ 1/(1 γ ). Thus, t suffces to show that 1/(1 γ ) > (αγ 1)/γ, that s, γ > (αγ 1)(1 γ ). Sce α s at most 2, (αγ 1)(1 γ ) < (2γ 1)(1 γ ) = γ + 2γ (1 γ ) 1. The result follows sce 2γ (1 γ ) 1. Our ext lemma shows that we ca cocetrate o the part of S where t < 1, whch wll allow us to use a represetato of the pot process volved. Lemma Assume that αγ > 1. The radom varable S s fte almost surely f ad oly f, j (g j x t γ ) ρ +1{t < 1} s. Proof. It suffces to show that (g j x t γ ) ρ +1{t 1} <, j

19 446 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) almost surely. A suffcet codto s that { : g x t γ > ; t 1} + { : g x t γ > ; t 1} < almost surely. Ths cardalty has a Posso dstrbuto wth fte mea sce for ay postve c, 1{cx t γ > ; t 1}α dx c α dt = xα+1 dt t γ s fte whe αγ > 1. Let (θ ) 1 be a sequece of depedet Beroull radom varables wth P {θ = 1} = 1/αγ. 1 Let (V ) 1 be a sequece of depedet radom varables, all beg uformly dstrbuted over [, 1], depedet of (θ ). Lemma Assume that αγ > 1. The, S s fte (resp. fte) almost surely f ad oly f 1 ρ/α h ρ (1/V γ )θ s almost surely fte (resp. fte). Proof. By cosderg separately the postve ad egatve x, we ca assume wthout loss of geeralty that p s 1. Let (U ) be a sequece of depedet radom varables, all uformly dstrbuted o [, 1]. Let also (ω ) be a sequece of depedet radom varables, all havg a expoetal dstrbuto wth mea 1. Furthermore, those sequeces are such that (θ ), (V ), (U ) ad (ω ) are depedet. Let W = ω ω. The radom measure 1 δ (W 1/α,U ) s a Posso process o [, ) [, 1] wth testy ν λ (recall that we assume that p s 1). It has the same dstrbuto as δ (x,t )1{x > ; t < 1}. 1 Thus, gve Lemma 4.5.3, S s fte f ad oly f (g j W 1/α U γ ) ρ +, j s fte almost surely. Ths latter sum s (g j W 1/α U γ ) ρ +1{U < g 1/γ W 1/αγ }., j Let θ = 1{U < g 1/γ W 1/αγ } ad let I = { : θ = 1}. Recall that for ay c [, 1] the dstrbuto of U gve U < c s that of cv. Thus, the codtoal dstrbuto of (U ) I gve (W ) ad I s that of V (g 1/γ W 1/αγ 1). Thus S has the same dstrbuto as I g j W 1/α V γ (g W 1/α 1) ρ θ +., j The strog law of large umbers esures that W almost surely as teds to fty. Thus, there are oly a fte umber of postve tegers such that g W 1/α > 1. Ths mples that S

20 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) s almost surely fte f ad oly f so s (g j W 1/α V γ g W 1/α ) ρ + θ =, j, j = W ρ/α (g j V γ g ) ρ + θ W ρ/α h ρ (V γ ) θ. (4.5.1) Set S(c) = ρ/α h ρ (V γ )1{U < cg 1/γ 1/αγ }. Sce W almost surely as teds to fty, (4.5.1) s almost surely fte f S(2) s; moreover, f S(1/2) s fte, so s (4.5.1). Thus, the followg clam shows that for S to be fte, t s ecessary ad suffcet that S(c) s fte for some c. Clam. S(c) s fte (resp. fte) almost surely for all postve c f ad oly f t s fte (resp. fte) for some postve c. Proof of the clam. Note that S(c) s odecreasg c. Hece, f t s fte for some c t s fte for ay c at most c. It s the suffcet to prove that f S(c) s fte, so s S(2c). But S(2c) S(c) = ρ/α h ρ (V γ cg 1/γ )1 1/αγ < U < 2 cg1/γ 1/αγ. For large eough so that 2cg 1/γ / 1/αγ < 1, the terval 1/γ (cg / 1/αγ )[1, 2] [, 1] has legth cg 1/γ / 1/αγ. I ths case, cg 1/γ 1 1/αγ < U < 2 cg1/γ 1/αγ has the same dstrbuto as 1{U < cg 1/γ 1/αγ }. Thus, there exsts a radom varable S(c) whch has the same dstrbuto as S(c) ad a almost surely fte radom varable Ω such that S(2c) S(c) Ω + S(c). It follows that f S(c) s fte, the so s S(2c). Ths proves our clam. Gve our clam ad the dscusso whch preceded t, the fteess of (4.5.1) s equvalet to that of S(g 1/γ ) =, j p/α h ρ (V γ )1{U < 1/αγ }. Ths proves the lemma sce (1{U < 1/αγ }) has the same dstrbuto as (θ ).

21 448 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Our ext step s to use the three seres theorem (see e.g. [7, Secto IX.9, Theorem 2]). It wll be useful to have ts statemet reproduced here whe the summads are oegatve radom varables. So, let (Y ) be a sequece of oegatve depedet radom varables. The three seres theorem asserts that for the sequece Y to coverge almost surely, t s ecessary ad suffcet that for ay postve s, P {Y > s} <, (4.5.2) EY 1{Y < s} <, (4.5.3) ad I our stuato, we set Var(Y 1{Y < s}) <. (4.5.4) Y = ρ/α h ρ (V γ )θ. We wll cosder these specfc Y whe we refer to (4.5.2) (4.5.4). Our ext lemma traslates (4.5.2). Lemma () If g ρ < the (4.5.2) holds. () If g ρ = the (4.5.2) s equvalet to (αγ 1)/γρ dh ρ (y) y 1/γ <. Proof. () Note that h ρ s at most g ρ. Thus, f ths seres coverges, Y h ρ ρ/α s less tha ay postve fxed s wheever s large eough. Ths mples (4.5.2). () Cosder a teger large eough so that h ρ (s ρ/α ) 1. We have P {Y > s} = P θ = 1; h ρ (V γ ) > s ρ/α = 1/αγ P V < h ρ (s ρ/α ) 1/γ = 1/αγ h ρ (s ρ/α ) 1/γ. Thus, (4.5.2) s equvalet to 1/αγ h ρ (s ρ/α ) 1/γ <. Sce h ρ s mootoe, ths s equvalet to x 1/αγ h ρ (sxρ/α ) 1/γ dx <. (4.5.5) The chage of varable y = h ρ (sxρ/α ) makes the tegral equal to a costat tmes h ρ (y) 1/ργ y 1/γ h ρ (y) (α/ρ) 1 dh ρ (y)(αγ 1)/γρ dh ρ (y) = y 1/γ. Our ext lemma traslates (4.5.3) ad (4.5.4).

22 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Lemma (4.5.3) ad (4.5.4) hold f ad oly f ρ > (αγ 1)/γ ad hρ (w)(αγ 1)/γρ w (1/γ ) 1 dw <. Proof. Note that EY 1{Y < s} = ρ/α E h ρ (V γ )θ 1{ ρ/α h ρ (V γ )θ < s}. Sce θ ad V are depedet ad θ has a Beroull dstrbuto wth mea 1/αγ, ths expectato s (ρ/α) (1/αγ ) E h ρ (V γ )1{h ρ (V γ ) < s ρ/α }. Thus, sce V are depedet ad equdstrbuted, EY 1{Y < s} 1 = E h ρ (V γ 1 ) (ρ/α) (1/αγ ) hρ 1 > (V γ α/ρ 1 ). (4.5.6) s 1 If (ρ/α) + (1/αγ ) 1, that s, ρ (αγ 1)/γ, the seres volved (4.5.6) dverges ad (4.5.3) does ot hold. Assume ow that ρ > (αγ 1)/γ. The seres volved (4.5.6) s of order c hρ (V γ ) s α ρ 1 ρ α 1 αγ Hece, (4.5.6) coverges or dverges accordg to whether. Eh ρ (V γ 1 ) 1+ α ρ 1 ρ α αγ 1 = Eh ρ (V γ 1 ) (αγ 1)/ργ (4.5.7) s fte or ot. Fteess of ths expectato s equvalet to the covergece of the tegral 1 h ρ (v γ ) (αγ 1)/γρ dv = 1 γ 1 h ρ (w) (αγ 1)/γρ w (1/γ ) 1 dw. Next, we cosder codto (4.5.4). Assume that (4.5.3) holds. The (4.5.4) s equvalet to the fteess of EY 2 1{Y < s} = (2ρ/α) (1/αγ ) Eh 2 γ hρ ρ (V )1 > (V γ α/ρ ) s. 1 1 The covergece of ths seres s equvalet to the fteess of Eh ρ (V γ ) 2+ α ρ 1 2ρ α 1 αγ whch s the same codto as (4.5.7). = Eh ρ (V γ ) (αγ 1)/γρ, The purpose of the ext lemma s to summarze what we have proved so far. Lemma The radom varable S s fte wth probablty or 1. It s almost surely fte f ad oly f all three followg codtos hold:

23 45 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) αγ 1 () ρ > γ > ; h () lm ρ (t) t = ; t ρ/(αγ 1) () h ρ (y)(αγ 1)/γρ dy <. y (1/γ )+1 Proof. Case ( g ρ ) summable. Assume ths case that g ρ s fte. Let us frst prove that the codtos are ecessary. Lemma shows that codto αγ > 1 s ecessary. Sce ( g ρ ) s summable, the fucto h ρ s bouded ad codtos () ad () are true. Lemma mples that codto () s true. Thus those codtos are ecessary, regardless whether S s fte or ot. Sce codtos () () are always true ths case, we eed to show that uder the addtoal codto (), S s fte. Codto () mples that αγ > 1. The, Lemmas () ad show that (4.5.2) (4.5.4) hold. The Lemma shows that S s almost surely fte. Case ( g ρ ) o-summable. Lemmas show that S s fte f ad oly f dh ρ (y)(αγ 1)/γρ y 1/γ < (4.5.8) ad ρ > (αγ 1)/γ, ad h ρ (w) (αγ 1)/γρ w (1/γ ) 1 dw <. Sce for ay fxed a t a dh ρ (y)(αγ 1)/γρ y 1/γ = c + h ρ(t) (αγ 1)/γρ t 1/γ + 1 γ ay two of the followg codtos mply the thrd oe: dh ρ (y)(αγ 1)/γρ h ρ (y)(αγ 1)/γρ y 1/γ dy <, y (1/γ )+1 dy <, h ρ (y)(αγ 1)/γρ lm y y 1/γ =. Ths mples the result. We cosder the fucto R(t) = { : g > g /t}. t a h ρ (y)(αγ 1)/γρ y 1+1/γ dy, (4.5.9) It vashes o [, 1] ad s odecreasg o (1, ); t dverges to fty at fty. Sce (g ) s regularly varyg of dex γ 1, a varato o the method used to prove Theorem [3] shows that R s regularly varyg of dex 1/(1 γ ). The fuctos h ρ ad R are related through the followg lemma. Lemma h ρ (v) = ρgρ v ρ Proof. We have h ρ (v) = R v 1+y y ρ 1 dy. 1{x < g (g /v)}ρx ρ 1 dx.

24 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) Sce the fuctos volved the summato ad tegrato are oegatve, Fub s theorem yelds h ρ (v) = = ρ 1{g > x + (g /v)}ρx ρ 1 dx g v R vx + g x ρ 1 dx. The result follows after the chage of varable x = g y/v. We wll use the followg lemma whch complemets Karamata s theorem for fractoal tegrals (see [8, Theorem C]; or [3, Chapter 1, Exercse 1]). Recall that the fractoal tegral of a fucto f s, whe t s well defed, ( η) f (v) = v f (x)(v x) η dx. Lemma Let f be a oegatve regularly varyg fucto of dex 1 defed o the oegatve half-le ad locally tegrable. The for ay oegatve η less tha 1, ( η) f (v) v η v f (x) dx as v teds to fty. I partcular, ( η) f s regularly varyg of dex η. Proof. Sce the statemet s tautologcal whe η vashes, we assume the proof that η s postve. It follows from Proposto 1.5.9a [3] that v f (x) dx v f (v) (4.5.1) as v teds to fty. Furthermore, usg the uform covergece theorem [3, Theorem 1.5.2], we obta that for ay postve ϵ less tha 1, v ϵv f (x) dx v f (v) 1 ϵ dy y as v teds to fty. Combed wth (4.5.1), t mples that for ay postve ϵ less tha 1, ϵv f (x) dx v f (x) dx (4.5.11) as v teds to fty. Let ϵ be a postve umber less tha 1. Sce f s regularly varyg ad oegatve, sce η s postve ad (4.5.11) holds, ϵv f (x)(v x) η dx v η (1 ϵ) η ϵv f (x) dx v v η (1 ϵ) η f (x) dx (4.5.12)

25 452 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) as v teds to fty. Smlarly, ϵv f (x)(v x) η dx v η ϵv v η v f (x) dx as v teds to fty. Next, usg the uform covergece theorem, f (vy) uformly y [ ϵ, 1 ], ad therefore v ϵv 1 f (x)(v x) η dx = v 1 η f (vy)(1 y) η dy ϵ v 1 η f (v) f (x) dx (4.5.13) 1 as v teds to fty. Therefore, usg (4.5.1), v v f (x)(v x) η dx = o f (x) dx ϵv ϵ y 1 (1 y) η dy f (v)/y as v teds to fty, (4.5.14) as v teds to fty. Combg (4.5.12) (4.5.14) ad usg the fact that ϵ s arbtrary, we obta the proper asymptotc equvalet for ( η) f. Proposto 1.5.9a [3] mples that ths asymptotc equvalet s regularly varyg of dex η. We ca ow coclude the proof of Proposto 1.7. We wll dstgush three cases accordg to the posto of ρ wth respect to 1/(1 γ ). () ρ < 1/(1 γ ). It follows from Lemma ad stadard regular varato theoretc argumets that f ρ < 1/(1 γ ), the h ρ (v) pg ρ R(v) v ρ y ρ 1 dy (1 + y) 1/(1 γ ) as v teds to fty. I partcular, h ρ s regularly varyg of postve dex 1/(1 γ ) ρ. Therefore, h αγ 1 γρ (αγ 1)/γρ ρ 1 1 γ ρ s regularly varyg of postve dex. Postvty of ths dex ad Karamata s theorem [3, Proposto 1.5.8], mply that codto () ad () of Lemma hold f αγ 1 1 γρ 1 γ ρ < 1 γ, that s ρ > ρ, ad fal wheever αγ 1 1 ρ 1 γ ρ > 1 γ, that s ρ < ρ. If ρ = ρ, the codtos () ad () Lemma are equvalet to the lmt ad tegral codtos Proposto 1.7().

26 Ph. Barbe, W.P. McCormck / Stochastc Processes ad ther Applcatos 122 (212) () ρ 1/(1 γ ) ad h ρ bouded. I ths case, sce 1/(1 γ ) = ρ αγ /(αγ 1), we have ρ > ρ ad codto () of Proposto 1.7 holds. I ths case, S s fte as we have see Lemma () ρ 1/(1 γ ) ad h ρ ubouded. I ths case, the oly possblty to have h ρ ubouded s to have ρ = 1/(1 γ ), for the seres g ρ coverges wheever ρ > 1/(1 γ ). I ths case, sce h ρ s mootoe, t teds to fty at fty. Clam. h ρ s slowly varyg. Proof of the clam. Chage of varable z = v/(1 + y) the tegral expresso for h ρ gve Lemma shows that h ρ (v) = ρgρ v ρ 1 v R(z) z ρ+1 (v z)ρ 1 dz. Note that the tegral s well defed sce R vashes o [, 1). Sce ρ = 1/(1 γ ), the fucto R(z)/z ρ+1 s regularly varyg of dex 1. The clam follows from Lemma Sce h ρ s slowly varyg, t grows to fty at a rate slower tha algebrac. Aga, codtos () ad () of Lemma are satsfed. Thus oly codto () s relevat. But ths codto () s always satsfed sce 1/(1 γ ) > (αγ 1)/γ. Thus S s fte. Moreover, ths stuato, ρ = 1/(1 γ ) > ρ ad assumpto () of Proposto 1.7 holds. Ackowledgmets We thak a aoymous referee ad Thomas Mkosch for dcatg the possblty of extedg the scope of the orgal verso of ths paper. We also thak a aoymous referee for detaled ad useful commets, ad for potg out some oversghts the orgal verso of ths paper. Refereces [1] Ph. Barbe, W.P. McCormck, Ivarace prcples for some FARIMA ad ostatoary lear processes the doma of attracto of a stable dstrbuto, Probab. Theory Related Felds (21) ( press). arxv: [2] Ph. Barbe, W.P. McCormck, Heavy-traffc approxmatos for fractoally tegrated radom walks the doma of attracto of a o-gaussa stable dstrbuto, Stochastc Process. Appl. 122 (212) [3] N.H. Bgham, C.M. Golde, J.L. Teugels, Regular Varato, secod ed., Cambrdge Uversty Press, [4] M. Braverma, T. Mkosch, G. Samorodtsky, Tal probabltes of subaddtve fuctoals of Lévy processes, A. Appl. Probab. 12 (22) [5] L. Devroye, Upper ad lower class sequeces for mmal uform spacgs, Z. Wahrschelchketstheor. Verwadte Geb. 61 (1982) [6] M. Dosker, Justfcato ad exteso of Doob s heurstc approach to the Kolmogorov Smrov theorems, A. Math. Statst. 23 (1952) [7] W. Feller, A Itroducto to Probablty Theory ad Its Applcatos, Wley, [8] J.L. Geluk, Π -regular varato, Proc. Amer. Math. Soc. 82 (1981) [9] J. Kefer, Iterated logarthm aalogues for sample quatles whe p, : Proc. Sxth Berkeley Sympos. o Math. Statst. ad Probab. vol. 1, 1972, pp [1] S.I. Resck, Heavy-Tal Pheomea, Probablstc ad Statstcal Modelg, Sprger, 27. [11] H. Robbs, A oe-sded cofdece terval for a ukow dstrbuto fucto, A. Math. Statst. 25 (1954) 49. [12] G.R. Shorack, J.A. Weller, Lear bouds o the emprcal dstrbuto fucto, A. Probab. 6 (1978) [13] W. Whtt, Stochastc-Process Lmts. A Itroducto to Stochastc-Process Lmts ad Ther Applcato to Queues, Sprger, 22.