Sharp large deviations for sums of bounded from above random variables

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1 Noame mauscript No. will be iserted by the editor Sharp large deviatios for sums of bouded from above radom variables Xiequa Fa Received: date / Accepted: date Abstract We show large deviatio epasios for sums of idepedet ad bouded from above radom variables. Our moderate deviatio epasios are similar to those of Cramér 1938, Bahadur ad Raga Rao 196, ad Sakhaeko I particular, our results eted Talagrad s iequality from bouded radom variables to radom variables havig fiite +δth momets, where δ, 1]. As a cosequece, we obtai a improvemet of Hoeffdig s iequality. Applicatios to liear regressio, self-ormalized large deviatios ad t-statistic are also discussed. Keywords sharp large deviatios Cramér large deviatios Talagrad s iequality Hoeffdig s iequality sums of idepedet radom variables Mathematics Subject Classificatio 6F1 6F5 6E15 6G5 1 Itroductio Let ξ i i 1 be a sequece of idepedet o-degeerate radom variables r.v.s satisfyig Eξ i =. The study of sharp large deviatios has a log history. May iterestig asymptotic epasios have bee established i Cramér [8], Bahadur ad Raga Rao [1], Petrov [17, 18], Saulis ad Statulevičius [1], Sakhaeko [], Nagaev [16], Bercu ad Rouault [5], Borovkov ad Mogulskii [6], Petrov ad Robiso [19] ad [1]. See also Grama ad Haeusler [13] ad [11] for martigales. For self-ormalized sums, we refer to Shao [3] ad Jig, Shao ad Wag [15], where the authors have established Cramér type large deviatios for self-ormalized r.v.s uder fiite +δth momets, where δ,1]. I this paper, we cosider the sharp large deviatios for sums of bouded from above r.v.s ξ i A for all i, where A is a positive costat. Without loss of geerality, we take A = 1, otherwise we cosider ξ i /A istead of ξ i. Thus ξ i 1 for all i. Let S = ξ i. Deote i = Eξ i ad = i. X. Fa Ceter for Applied Mathematics, Tiaji Uiversity, 37 Tiaji, Chia; Regularity Team, INRIA Saclay, Palaiseau, 911, Frace faiequa@hotmail.com

2 Xiequa Fa The celebrated Beett iequality [] states that: If ξ i 1 for all 1 i, the for all >, + PS B, := e. 1 However, Beett s iequality is ot tight eough. Oe of the improvemets o Beett s iequality is Hoeffdig s iequality cf..8 of [14], which states that for all >, PS H, + } + := 1 + } B,, 3 where ad hereafter by covetio = 1 applied whe =. Cosiderig the followig distributio law Pη i = 1 = / 1+ / ad Pη i = / = 1 1+ /, 4 Hoeffdig showed that is the best that ca be obtaied from the followig epoetial Markov iequality PS if λ EeλS,. Ideed, it is easy to see that Eη i =,η i 1, Eη i = / ad if Eep λ λ } η i = H, for all. Notice that lim PS > = 1 Φ ad lim H, = e /, where Φ = 1 π e t dt is the stadard ormal distributio fuctio. The cetral limit theorem CLT suggests that Hoeffdig s iequality ca be substatially refied by addig a missig factor Θ as, where Θ = The factor Θ satisfies 1 Φ ep } = O 1, Θ,, 6 π1+ π1+ ad πθ is kow as Mill s ratio. For sums of bouded r.v.s B ξ i 1 for some costat B 1 ad all 1 i, Talagrad [4] proved the followig iequalities: For all C B, PS if Θ+C B Θ+C B λ EeλS 7 H,, 8

3 Sharp large deviatios for sums 3 wherec > isaabsolutecostat.seealsosakhaeko[]foraresultsimilarto7butθ isreplacedbyθα, whereα = sup λ [λ logee λs ]} 1/.By6,Talagrad siequality 1 8 improves Hoeffdig s iequality by addig a factor Θ[1 + o1] of order 1+ i the rage = o B as B. I the i.i.d. case, this rage reduces to = o,. I this paper, we eted Talagrad s iequality 7 from bouded r.v.s to r.v.s havig fiite +δth momets, δ,1]. I particular, we improve Talagrad s iequality to a equality, which will imply simple large deviatio epasios. Moreover, a improvemet of Hoeffdig s iequality uder fiite +δth momets is also give. The paper is orgaized as follows. I Sectio, we preset the etesio of Talagrad s iequality. I Sectio 3, we apply our result to liear regressio. I Sectio 4, we give two lower bouds for self-ormalized moderate deviatios ad t-statistic. I Sectio 5, we prepare some auiliary results. I Sectios 6 ad 7, we prove the mai theorems. A etesio of Talagrad s iequality Throughout the paper, we make use of the followig otatios: a b = mia,b}, a b = maa,b}, a + = a, ad θ stads for some values satisfyig θ 1. Moreover, we deote C ad C δ, probably supplied with some idices, a geeric positive absolute costat ad a geeric positive costat depedig oly o δ, respectively. Our mai result is the followig theorem, which eteds Talagrad s iequality 7 from bouded r.v.s to r.v.s havig fiite +δth momets, δ,1]. Theorem 1 Assume that ξ i 1, ad that there eist two costats B 1 ad δ,1] such that The for all C δ B, PS = E ξ i +δ B δ Eξi, i 1. 9 B δ Θ+θC if λ EeλS. 1 I particular, i the i.i.d.case, it implies that for all = o δ/, as. PS = 1+o1 11 Θif λ EeλS By ispectig the proof of Theorem 1, we ca see that Theorem 1 holds true for C δ = 3 61/δ ad C = To show the tightess of equality 1, let S = ε ε be the sums of idepedet Rademacher r.v.s, i.e. Pε i = ±1 = 1 for all i. We display the tail probabilities ad the simulatio of PS R, = Θif λ Ee = PS λs ΘH, i Figure 1, which shows that R, is very close to 1 for large s. I the followig corollary, we give a improvemet o Hoeffdig s iequality.

4 4 Xiequa Fa Approimatio for = 49 Compariso for = 49 Probabilities PS ΘH, Ratios R, Approimatio for = 1 Compariso for = 1 Probabilities PS ΘH, Ratios R, Approimatio for = 1 Compariso for = 1 Probabilities PS ΘH, Ratios Fig. 1 Tail probabilities ad ratios R, are displayed as a fuctio of ad various. Corollary 1 Assume coditio of Theorem 1. The for all C δ B, B δ PS Θ+C 1 H,. 1 It is clear that iequality 1 improves Hoeffdig s boud H, by addig a missig factor Θ+C B δ 1. By 6, this factor is of order of Θ[1+o1] i the rage = o /B δ as B/. I the i.i.d. case, this rage reduces to = o δ/ as. For r.v.s ξ i without momets of order larger tha, some improvemets of Hoeffdig s iequality ca be foud i Betkus [3] ad Betkus, Kalosha ad va Zuijle [4]. See also Pielis [] for a improvemets of Beett-Hoeffdig s iequality 3 which is larger tha

5 Sharp large deviatios for sums 5 Hoeffdig s iequality. I particular, whe ξ i 1 for all 1 i, Betkus [3] showed that PS e Po η i, 13 whereη i arei.i.d.withdistributio4adp o η i isthelog-cocavehullofp η i,i.e.p o isthemiimumlog-cocavefuctiosuchthatp o P.Applyig1toP η i with B = ma1, } ad δ = 1, we have for all 1 C mi,}, 1 P η i = Θ+O1ma, } H,. 14 By the iequalities 13 ad 14, we fid that 1 refies Betkus costat e to 1+o1 for all = o ma 1 δ, } as. Iequality 1 implies the followig Cramér type large deviatios. Corollary Assume coditio of Theorem 1. The for all where ˇ = 1+ 3 PS ad satisfies C δ B, 1 Φˇ [ B ] δ 1+C1+ ˇ, 15 ˇ = 1 6 +o as. I particular, i the i.i.d. case, it implies that for all = o δ/, [ ] PS 1 Φˇ 1+o1. 16 The iterestig feature of the boud 16 is that it closely recoversthe shape of the stadard ormal tail for all = o δ/ as. 3 Applicatio to liear regressio The liear regressio model is give by X k = θφ k +ε k, k 1, 17 where X k,φ k ad ε k are, respectively, the respose variable, the positive covariate ad the oise. Let ε k k 1 be a sequece of i.i.d. radom variables, with fiite variace Eε k = 1 >. Our iterest is to estimate the ukow parameter θ, based o the radom variables X k k 1 ad φ k k 1. The well-kow least squares estimator θ is give by θ = k=1 φ kx k. 18 k=1 φ k Cosider the self-ormalized approimatio θ θ Σk=1 φ k. I the real-world applicatios, for istace cosiderig the impact of the footprit size φ k o the height X k, it is plausible that a φ k b for two positive absolute costats a ad b. If X k, the we also have ε k = X k θϕ k θb c for a positive absolute costat c. By Theorem 1 ad Corollary 1, we have the followig result.

6 6 Xiequa Fa Theorem Assume that there eist three positive absolute costats a, b ad c such that for all k 1, a φ k b, ε k c. We also assume that E ε k +δ < for a absolute costat δ,1]. The for all = o, P θ θ Σk=1 φ k 1 = Θ+θC 1 if Eep φ k ε } k λ δ/ λ + 1 k=1 φ k Θ+C 1 δ/ Proof. From 17 ad 18, it is easy to see that Set H, a1. bc θ θ Σ k=1 φk = φ k ε k. 19 k=1 φ k ξ i = φ kε k a bc. k=1 φ k The it is easy to verify that ξ i 1, Eξi = a 1 a b c ad θ θ Σk=1 φ k bc k=1 = ξ i. By Theorem 1 ad Corollary 1, it follows that for all = o, P θ θ Σk=1 φ k 1 = Θ+θC 1 if Eep a1 } λ ξ δ/ i λ bc = Θ+θC 1 if Eep φ k ε } k λ δ/ λ + 1 k=1 φ k which completes the proof of theorem. Θ+C 1 δ/ H, a1, bc 4 Applicatios to self-ormalized deviatios ad t-statistic Limit theorems for self-ormalized sums S /V, V = ξ i, put a totally ew couteace o classical limit theorems. It is well kow that self-ormalized limit theorems require much fewer momet coditios tha that of ormalized limit theorems. For eample, uder fiite +δth momets, δ,1], Jig, Shao ad Wag [15] showed that PS /V > 1 Φ uiformly for all = omiε 1, κ 1 }, where ε δ = E ξ i / +δ = ep O 1 } 1+ +δ ε δ ad κ = ma 1 i Eξ i/.

7 Sharp large deviatios for sums 7 The proof of lower boud for self-ormalized Cramér type large deviatios is based o the followig observatio: For ay,λ >, } S /V S +λ V } = λ } ζ i λ, where ζ i λ = λξ i 1 λ ξi. Thus P S /V P ζ iλ. Notice that λξi 1 λ ξi i 1 is also a sequece of idepedet radomvariablesadsatisfiesλξ i 1 λ ξi 1.Byaargumetsimilartothe proof of Theorem 1, we have the followig lower boud o the tail probabilities of self-ormalized sums. Theorem 3 Assume that there eist two costats B > ad δ,1] such that The for all = o B, where P S /V P E ξ i +δ B δ Eξ i, i 1. = ep Ψ λ = ζ i λ +Ψ λ } B δ Θ+θC, 1 logee ζiλ ad λ is defied by the followig equatio Eζ i λe ζiλ Ee ζiλ =. Moreover, it holds for all = o B, Ψ λ = O1 +δ B δ, 3 where O1 is bouded by a absolute costat. The self-ormalized sums are closely related to Studet s t-statistic. Studet s t-statistic T is defied by the followig formula T = ξ /, where ξ = S It is kow that for all, P T ad = = P S /V ξ i ξ /. See Efro [9]. By the last equality, oce we have a estimatio for the tail probabilities of self-ormalized sums S /V, we have a similar estimatio for the tail probabilities of T. So Theorem 3 implies the followig lower boud o tail probabilities of T.

8 8 Xiequa Fa Theorem 4 Assume the coditio of Theorem 3. The for all = o B, P T ep B + +Ψ λ} Θ +θc / where Ψ λ is defied by ad λ is defied by the followig equatio Eζ i λe ζiλ Ee ζiλ = + 1. δ, 5 Auiliary Results Cosider the positive radom variable Z λ = e λξi Eeλξi, λ, so that EZ λ = 1 the Esscher trasformatio. Itroduce the cojugate probability measure P λ defied by dp λ = Z λdp. 4 Deote by E λ the epectatio with respect to P λ. Settig ad we obtai the followig decompositio: where B k λ = b i λ = E λ ξ i = Eξ ie λξi Ee λξi, i = 1,...,, η i λ = ξ i b i λ, i = 1,...,, S k = B k λ+y k λ, k = 1,...,, 5 k b i λ ad Y k λ = k η i λ. I the proof of our mai result, we shall eed a two-sided boud of B λ. To this ed, we eed some techical lemmas. For a radom variable bouded from above, the followig iequality is well-kow. Lemma 1 Assume ξ 1. Deote by the stadard variace of ξ. The for all λ, where Ee λξ Beλ,, Beλ,t = t 1+t epλ}+ 1 1+t ep λt}. A proof of the iequality ca be foud i Beett []. This iequality is sharp, ad it attais to equality whe ξ has the distributio law: Pξ = 1 = 1+ ad Pξ = = 1 1+.

9 Sharp large deviatios for sums 9 Lemma Assume E ξ +δ B δ Eξ for some costats B ad δ,1]. The Eξ B. 6 Proof. Usig Jese s iequality, we deduce that Eξ +δ/ E ξ +δ B δ Eξ, which implies 6. Lemma 3 Assume E ξ +δ B δ Eξ for some costats B ad δ,1]. The for all λ, Eξ e λξ 1 B δ λ δ Eξ. 7 Proof. Usig the followig iequality t e t t 1 t δ,t R, we deduce that for all λ, Eλξ e λξ Eλξ E λξ +δ λ Eξ λb δ Eξ, 8 which gives the desired iequality. I the followig lemma, we give a two-sided boud for B λ. Lemma 4 Assume ξ i 1 for all i. The for all λ, B λ e λ 1. If ξ i satisfies E ξ i +δ B δ Eξi for some costats B ad δ,1] for all i, the for all λ, B λ 1 Bδ λ δ λ e λ. 1+δ Proof. By Jese s iequality, we have for all λ, Ee λξi e λeξi = 1. Notice that Eξ i e λξi = Eξ i e λξi 1 for λ. The by the fact ξ i 1, we obtai the upper boud of B λ : For all λ, B λ Eξ i e λξi = λ i et dt = e λ 1. λ Eξ ie tξi dt If ξ i satisfies E ξ i +δ B δ Eξi, by Lemma 3, it follows that for all λ, λ Eξ i e λξi = Eξie tξi dt λ 1 B δ t δ dt = 1 Bδ λ δ λ. 1+δ Therefore, we get the lower boud of B λ: For all λ, Eξ i e λξi B λ = 1 Bδ λ δ λ e λ, Ee λξi 1+δ Eξ i

10 1 Xiequa Fa which completes the proof of Lemma 4. Net, we give a upper boud for the cumulat fuctio Ψ λ = logee λξi, λ. 9 Lemma 5 Assume ξ i 1 for all i. The for all λ, 1 Ψ λ log 1+ / ep λ / } + / 1+ / epλ}. Proof. Sice the fuctio fλ,t = log 1 1+t ep λt}+ t 1+t epλ}, λ,t, has a egative secod derivative i t >, the for ay fied λ, fλ,t is cove i t. Therefore, by Lemma 1 ad Jese s iequality, we get for all λ, Ψ λ fλ,i fλ, / 1 = log 1+ / ep λ / } + / 1+ / epλ}. This completes the proof of Lemma 5. Deote the variace of Y λ by λ = E λ Yλ, λ. By the relatio betwee E ad E λ, it is obvious that Eξ λ = i e λξi Eξ ie λξi Ee λξi Ee λξi, λ. The followig lemma gives some estimatios of λ. Lemma 6 Assume ξ i 1, ad that E ξ i +δ B δ Eξi i. The for all λ, for some costats B, δ,1] ad all e λ 1 B δ λ δ B e λ 1 λ eλ. 3 Moreover, if B 1, it holds λ 1 5B δ λ δ Proof. Sice Ee λξi 1,λ, ad ξ i 1, we get for all λ, λ Eξi Eξ eλξi i eλ = e λ. This gives the upper boud of λ. Net, we cosider the lower boud of λ. It is easy to see that for all λ, Eξ i e λξi = λ Eξ ie tξi dt λ e t Eξ idt = e λ 1 Eξ i, 3

11 Sharp large deviatios for sums 11 Usig Lemma 3 ad 3, we obtai for all λ, Eξi Eξ λ eλξi i e λξi e λ 1 Bδ λ δ e λ 1 Eξ i e λ e λ 1 B δ λ δ B e λ 1, which gives the first lower boud of λ. Notig that λ ad B 1, by a simple calculatio, we obtai 31. Forthe radomvariabley λ, λ,wehavethefollowigresultothe rateofcovergece to the stadard ormal law. Lemma 7 Assume that ξ i 1, ad that E ξ i +δ B δ Eξi for some costats B, δ,1] ad all i. The for all λ, sup P Y λ λ y R λ y Φy +δ Cδ e λ +δ λ E ξ i +δ. Proof. Notice that Y λ = η iλ is the sum of idepedet r.v.s η i λ ad E λ η i λ =. Usig the well-kow rate of covergece i the cetral limit theorem cf. e.g. [17], p. 115, we get for all λ, sup P λ y R Y λ λ y Φy Cδ +δ λ E λ η i +δ. 33 Usig the iequality a+b 1+q q a 1+q +b 1+q for a,b,q, we deduce that for all λ, E λ η i +δ 1+δ E λ ξ i +δ + E λ ξ i +δ +δ E λ ξ i +δ +δ E ξ i +δ e λξi +δ e λ E ξ i +δ. Therefore, we obtai for all λ, sup P Y λ λ λ y Φy +δ Cδ e λ +δ λ y R E ξ i +δ. This completes the proof of Lemma 7. We are ow ready to prove the mai techical result of this sectio. Theorem 5 Assume that ξ i 1, ad that E ξ i +δ B δ Eξi for some costats B ad δ,1] for all i. For a, if there eists a positive λ such that Ψ λ =, the PS = Θ λλ +θε if λ EeλS, 34 where ε = 3+δ Cδ e λ +δ λ E ξ i +δ.

12 1 Xiequa Fa Proof. Accordig to the defiitio of the cojugate probability measure cf. 4, we have the followig represetatio of PS : For give,λ, PS = E λ Z λ 1 1 S } = E λ e λs+ψλ 1 S } = E λ e λ+ψλ λyλ λbλ+λ 1 Yλ+B λ } = e λ+ψλ E λ e λ[yλ+bλ ] 1 Yλ+B λ }. Settig U λ = λy λ+b λ, we get PS = e λ+ψλ e t P λ < U λ tdt. 35 For a, if there eists a λ = λ such that Ψ λ =, the the epoetial fuctio e λ+ψλ i 35 attais its miimum at λ = λ. Sice B λ = Ψ λ =, we have U λ = λy λ ad Usig Lemma 7, we deduce that e λ+ψλ = if λ e λ+ψλ = if λ EeλS. 36 e t P λ < U λ tdt = = = e λyλ P λ < U λ λyλ λλdy e λyλ P < N,1 yλλdy +θε e λyλ dφy+θε = Θ λλ +θε, 37 where N,1 stads for the stadard ormal r.v. Therefore, from 35 ad 36, it follows that PS = Θ λλ +θε if λ EeλS. This completes the proof of Theorem 5. 6 Proof of Theorem 1 I the spirit of Talagrad [4], we would like to make use of Θ to approimate Θλλ i Theorem 5. The proof of Theorem 1 is a cotiuatio of the proof of Theorem 5. Proof of Theorem 1. Sice Θ 1 π, we deduce that 1 Θ λλ Θ 1 λλ π λ λ Usig Lemma 4, we have for all λ 1 B, 1 Bδ λ δ e λ λ B λ = e λ δ

13 Sharp large deviatios for sums 13 By the estimatio of λ i Lemma 6, it follows that for all λb 6 1/δ, ad [ e λλ λ 1 λ 1 5B δ λ δ e λ 1 Bδ λ δ e λ] λ 1+δ [ λ 1+ 1 λbeλb 1 4B δ λ δ 3 λb λ B e λb +λ δ B δ e λb] [ λ 4B δ λ δ + 1 e 1 6 λb 3 λb e 1 1 λb +λ δ B δ ] 4.6λB δ λ 4 λ λ λ 1 5B δ λ δ 1 5 λ 6 λ e λ 1 Bδ λ δ 1+δ e λ 1 6 λ. 41 Hece, iequality 38 implies that for all λ 1 6 1/δ B, Θ λλ Θ 7.6 B πλ 1 δ δ1λ 6} λ 1+δ B π δ1λ< 6} 7.6 π B δ. 4 Therefore, by Lemma 6 ad coditio 9, it is easy to see that for all λ 1 6 1/δ B, e λ +δ λ E ξ i +δ e 1/6 E ξ i +δ B δ. 5/6 1+δ/ +δ By 39 ad the iequality e 1 for all, it follows that ad 1 B δ λ δ λ λ 1 6 1/δ B for all B 3 6 1/δ. 44 Combiig 34, 4 ad 43 together, we have for all 3 6 1/δ B, PS = Θ+θ 7.6 B δ C δ if π λ EeλS, where C δ is the smallest costat such that 33 holds. By Theorem.1 of Che ad Shao [7], we have C δ 4.1. Thus 7.6 π C δ This completes the proof of Theorem 1.

14 14 Xiequa Fa Proof of Corollary 1. Usig 36 ad Lemma 5, we get for all, if λ EeλS = if λ Ee λ+ψλ if ep λ +log λ = H, / ep λ / } } + / 1+ / epλ} 45 Notice that if λ Ee λs Ee S = 1. Thus, the desired iequality follows from Theorem 1 ad 45. Proof of Corollary. Sice Θ is decreasig i, we deduce that Θ Θˇ where ˇ =. Notice that Hoeffdig s boud is less tha Berstei s boud, i.e H, ep ˇ } cf. Remark.1 of [1]. Therefore, from 1, we have for all PS Usig 6, we obtai for all C δ B, PS B δ Θˇ+C ep = 1 Φˇ+C ˇ } B δep ˇ }. 1 Φˇ [ B ] δ 1+C1+ ˇ. C δ B, This completes the proof of Corollary. 7 Proof of Theorem 3 Cosider the positive radom variable H λ = e ζiλ Eeζiλ, λ, so that EH λ = 1. Itroduce the followig ew cojugate probability measure P λ defied by dp λ = H λdp. 46 I this sectio, deote by E λ the epectatio with respect to P λ defied by 46. Accordig to 46, we have the followig represetatio: For all = o B, ] P ζ i λ = E λ [H λ 1 1 ζiλ } = E λ [ep ] η i λ B λ+ψ λ }1 ζiλ }

15 Sharp large deviatios for sums 15 where ad = E λ [ep = E λ [ep B λ = ] η i λ B λ+ψ λ }1 ζiλ } ] η i λ B λ+ψ λ }1 ηiλ Bλ},47 η i λ = ζ i λ E λ ζ i λ E λ ζ i λ = Eζ i λe ζiλ Ee ζiλ Let λ = λ be the smallest positive solutio of the followig equatio Recall From 47, we obtai P ζ i λ = ep = ep B λ =. Y λ = η i λ. } +Ψ λ E λ [e Yλ 1 ] Yλ } } +Ψ λ e y P λ Y λ ydy. 48 Set F y = P λ Y λ y. Recall Cδ 4.1 see Theorem.1 of Che ad Shao [7]. By a argumet similar to the proof of Lemma 7, we have for all = o B, B δ. sup F y Φy/λ C 1 y R Hece, for all = o B, e y P λ Y λ ydy e y P N,1 y/λdy B δ, 1 C where N,1 is the stadard ormal r.v. By a simple calculatio, it follows that e y P N,1 y/λdy = Θλ. From 48, we have for all = o B, P ζ i λ = ep +Ψ λ } B Θλ+θC δ. 49 Net, we would like to substitute for λ i the item Θλ. By a argumet similar to the proof of 4, we get for all = o B, B δ. Θλ Θ C 3 5

16 16 Xiequa Fa So we have for all = o B, } P ζ i λ = ep B δ +Ψ λ Θ+θC, 51 which gives the desired epasio of tail probabilities. I the sequel we shall give a estimatio for Ψ λ. To this ed, we eed the followig useful lemma cf. Lemma 6. of Jig, Shao ad Wag [15]. Lemma 8 Let X be a radom variable with EX = ad EX <. For λ >, let ζ = λx 1 λx. The for λ >, Ee ζ = 1+O1ε λ, 5 Eζe ζ = 1 λ EX +O1ε λ, 53 where ε λ = λ E[X 1 λx >1} ]+λ 3 E[ X 3 1 λx 1} ]. Notice that ε λ E λx +δ = λ +δ E X +δ. Sice E ξ i +δ B δ Eξi, by Lemmas ad 8, for ay λ = ob 1, Ψ λ = O1 λ +δ E ξ i +δ = O1λ +δ B δ 54 ad Eζ i λe ζiλ = 1 Ee ζiλ λ +O1λ +δ B δ. Recall that λ = λ > is the smallest positive solutio of the followig equatio Eζ i λe ζiλ Ee ζiλ =, we have for all = o B, Thus, from 54, it holds λ = O1. Ψ λ = O1 +δ B δ, which gives the estimatio of Ψ λ. This completes the proof of Theorem 3 Ackowledgemets We would like to thak the aoymous referee ad editor for their helpful commets ad suggestios. The work has bee partially supported by the Natioal Natural Sciece Foudatio of Chia Grat os ad

17 Sharp large deviatios for sums 17 Refereces 1. Bahadur R, Raga Rao R. O deviatios of the sample mea. A Math Statist, 196, 31: Beett G. Probability iequalities for sum of idepedet radom variables. J Amer Statist Assoc, 196, 57: Betkus V. O Hoeffdig s iequality. A Probab, 4, 3: Betkus V, Kalosha N, va Zuijle M. O domiatio of tail probabilities of supermartigales: eplicit bouds. Lithuaia Math J, 6, 46: Bercu B, Rouault A. Sharp large deviatios for the Orstei-Uhlebeck process. Theory Probab Appl, 6, 46: Borovkov A A, Mogulskii A A. O large ad superlarge deviatios of sums of idepedet radom vectors uder Cramer s coditio. Theory Probab Appl, 7, 51: Che L H Y, Shao Q M. A o-uiform BerryCEssee boud via Stei s method. Probab Theory Relat Fields, 1, 1: Cramér H. Sur u ouveau théorème-limite de la théorie des probabilités. Actualite s Sci Idust, 1938, 736: Efro B. Studet s t-test uder symmetry coditios. J Amer Statist Assoc, 1969, 6438: Fa X, Grama I, Liu Q. Hoeffdig s iequality for supermartigales. Stochastic Process Appl, 1, 1: Fa X, Grama I, Liu Q. Cramér large deviatio epasios for martigales uder Berstei s coditio. Stochastic Process Appl, 13, 13: Fa X, Grama I, Liu, Q. Sharp large deviatio results for sums of idepedet radom variables. Sci Chia Math, 15, 58: Grama I, Haeusler E. Large deviatios for martigales via Cramer s method. Stochastic Process Appl,, 85: Hoeffdig W. Probability iequalities for sums of bouded radom variables. J Amer Statist Assoc, 1963, 58: Jig B Y, Shao Q M, Wag Q. Self-ormalized Cramér-type large deviatios for idepedet radom variables. A Probab, 3, 31: Nagaev S V. Lower bouds for the probabilities of large deviatios of sums of idepedet radom variables. Theory Probab Appl,, 46: Petrov V V. Sums of Idepedet Radom Variables. Berli: Spriger-Verlag, Petrov V V. Limit Theorems of Probability Theory. Oford: Oford Uiversity Press, Petrov V V, Robiso J. Large deviatios for sums of idepedet o idetically distributed radom variables. Comm Statist Theory Methods, 8, 37: Pielis I. O the Beett-Hoeffdig iequality. A Ist H Poicaré Probab Statist, 14, 5: Saulis L, Statulevičius V A. Limit theorems for large deviatios. Netherlads: Spriger, Sakhaeko A I. Berry-Essee type bouds for large deviatio probabilities. Siberia Math J, 1991, 3: Shao Q M. A Cramér type large deviatio result for Studet s t-statistic. J Theoret Probab, 1999, 1: Talagrad M. The missig factor i Hoeffdig s iequalities. A Ist H Poicaré Probab Statist, 1995, 31: 689-7