A note to the convergence rates in precise asymptotics

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1 He Journal of Inequaliies and Alicaions 203, 203:378 h:// R E S E A R C H Oen Access A noe o he convergence raes in recise asymoics Jianjun He * * Corresondence: hejj@cjlu.edu.cn Dearmen of Mahemaics, China Jiliang Universiy, Hangzhou, 3008, China Absrac Le {X, X n, n } be a sequence of i.i.d. random variables wih zero mean. Se S n = n k= X k, EX 2 = σ 2 >0,andλ r, ɛ= nr/ 2 P S n n / ɛ. In his aer, he auhor discusses he rae of aroximaion of r E N 2r /2 by ɛ 2r /2 λ r, ɛ under suiable momen condiions, where N is normal wih zero mean and variance σ 2 > 0, which imroves he resuls of Gu and Seinebach J. Mah. Anal. Al. 390:-4, 202 and exends he work He and Xie Aca Mah. Al. Sin. 29:79-86, 203. Secially, for he case r =2and = β+, β >, he auhor discusses he rae of 2 σ aroximaion of 2 2β+ by ɛ2 λ 2,/β+ ɛ under he condiion EX 2 I X > =O δ l for some δ >0,wherel is a slowly varying funcion a infiniy. MSC: 60F5; 60G50 Keywords: convergence rae; recise asymoics; slowly varying funcion Inroducion Le {X, X n, n } be a sequence of i.i.d. random variables. Se S n = n k= X k and λ r, ɛ= nr/ 2 P S n n / ɛ. Heyde [rovedha lim ɛ 0 ɛ2 λ 2, ɛ=σ 2, whenever EX =0andEX 2 = σ 2 <. Klesov[2 sudied he rae of he aroximaion of σ 2 by ɛ 2 λ 2, ɛ under he condiion E X 3 <. HeandXie[3 imroved he resuls of Klesov [2. Gu and Seinebach [4 exended he resuls of Klesov [2 and obained he following Theorem A. Gu and Seinebach [5 sudied he general idea of roving recise asymoics. Theorem A Le {X, X n, n } be a sequence of i.i.d.random variables wih zero mean and 0< <2,r 2. If EX 2 = σ 2 >0and E X q < for some r < q 3, hen ɛ 2r /2 λ r, ɛ r E N 2r /2 = o ɛ q 2r q 2. 2 If EX 2 = σ 2 >0and E X q < for some q 3 wih q > 2r 3 2, hen ɛ 2r /2 λ r, ɛ r E N 2r /2 = o 2r ɛ 2 +2q q, where N is normal wih mean 0 and variance σ 2 > He; licensee Sringer. This is an Oen Access aricle disribued under he erms of he Creaive Commons Aribuion License h://creaivecommons.org/licenses/by/2.0, which ermis unresriced use, disribuion, and reroducion in any medium, rovided he original work is roerly cied.

2 HeJournal of Inequaliies and Alicaions 203, 203:378 Page 2 of h:// The urose of his aer is o srenghen Theorem A andexendheheoremofheand Xie [3 under suiable momen condiions. In addiion, we shall discuss he rae a which ɛ 2 λ 2,/β+ ɛconvergeso σ 2 2β+ under he condiion T=O δ l for some δ >0,where T =EX 2 I X >, l is a slowly varying funcion a infiniy. Throughou his aer, C reresens a osiive consan, hough is value may change from one aearance o he nex, and [x denoes he ineger ar of x. x is he sandard normal disribuion funcion, x= x 2π e 2 /2 d, ϕx= x. 2 Main resuls From Gu and Seinebach [6, i is easy o obain he followinglemma. Lemma 2. Le {X, X n, n } be a sequence of i.i.d. normal disribuion random variables wih zero mean and variance σ 2 >0.Se 0< <2and r 2, hen ɛ 2r /2 n r/ 2 P S n n / ɛ Oɛ 2r /2, r <3, = Oɛ 4/2, r 3. r E N 2r /2 2. Lemma 2.2 Bingham e al. [7 Le l beaslowlyvaryingfuncion. Wehave for any η >0, lim η l=, lim η l=0; 2 if 0<δ <,hen a s δ ls ds δ δ l, ; 3 if δ >,hen s δ ls ds δ δ l, ; 4 if δ =,hen L= ls s ds, m= ls a ds are slowly varying funcions; and s l lim L =0, lim l m =0. Theorem 2. Le {X, X n, n } be a sequence of i.i.d.random variables wih zero mean and 0< <2,r 2. If EX 2 = σ 2 >0and E X 3 < for some r <3,hen ɛ 2r /2 λ r, ɛ Oɛ 2r /2, 2 r < 3 2, r E N 2r /2 = Oɛ /2 log ɛ, r = 3 2, Oɛ /2 3, 2 < r <3. 2.2

3 HeJournal of Inequaliies and Alicaions 203, 203:378 Page 3 of h:// 2 If EX 2 = σ 2 >0and E X 2+δ < for some 0<δ <,r <2+δ, hen ɛ 2r /2 λ r, ɛ r E N 2r /2 Oɛ 2r /2, 2 r <+δ/2, = Oɛ δ/2 log ɛ, r =+δ/2, oɛ δ/2, + δ/2 < r <2+δ If EX 2 = σ 2 >0and E X q < for some q 3 wih q > 2r 3 2, hen ɛ 2r /2 λ r, ɛ Oɛ 2r /2, 2 r < 3 2, r E N 2r /2 = Oɛ /2 log ɛ, r =3/2, Oɛ /2, r >3/2, 2.4 where N is normal wih mean 0 and variance σ 2 >0. Remark 2. Clearly, Theorem and Theorem 2 in He and Xie [3 aresecialcasesof Theorem 2.,byakingr =2and =. Remark 2.2 If 0 < <2,r 2, we have min 2r 2, δ 2 > δr for r <2+δ = q 3 2+δ 2 and min 2r 2, 2 > 2r 2r 3 for some q 3wihq >. So, he resuls of Theorem 2. are sronger han hose of Theorem 2 +2q q 2 A. Theorem 2.2 Le {X, X n ; n } be a sequence of i.i.d random variables wih zero mean, and le T=O δ l for some δ >0,where l is a slowly varying funcion a infiniy. Se EX 2 = σ 2 >0and β > 2. If δ >,hen ɛ 2 λ 2,/β+ ɛ σ 2 Oɛ 2, 2 < β < 4, 2β + = Oɛ 2 log ɛ, β = 4, 2.5 Oɛ /2β+, β > 4. 2 If 0<δ <,hen ɛ 2 λ 2,/β+ ɛ σ 2 2β + = Oɛ 2, 2 < β < 2 + δ 4, Oɛ δ/2β+ lɛ /2β+, β 2 + δ If δ =,hen ɛ 2 λ 2,/β+ ɛ σ 2 Oɛ 2 + ɛ 2 ɛ /2β+ 2β + = Oɛ 2 + ɛ 5 Oɛ /2β+ + ɛ 2β+3/2β+ l d, 2 < β < 4, l d log ɛ, β = 4, l d, β > Remark 2.3 For r =2, =.Ifl =, hen he resul of Theorem 2.2 is weaker han β+ ha of Theorem 2. for 0 < δ <,β 2 + δ, and weaker han ha of Theorem 2. for 4

4 HeJournal of Inequaliies and Alicaions 203, 203:378 Page 4 of h:// δ =. Bu he condiion T=O δ is weaker han he condiion E X 2+δ <.Ifl 0 as, hen he resul of Theorem 2.2 ishesameashaoftheorem2. for 0 < δ <. Remark 2.4 For δ > 0, he condiion E X 2+δ < is neiher sufficien nor necessary for he condiion T=O δ l. Here are some suiable examles. C+δ ln x Examle Le X be a random variable wih densiy f x= I x > e, where C is x 3+δ ln 2 x a normalizing consan, and 0 < δ <,henex =0andT = C I > e, l = δ ln ln is a slowly varying funcion a infiniy. Bu E X 2+δ = C x >e dx =. +δ ln x x ln 2 x Examle 2 Le X be a random variable wih densiy f x = Cδ ln2 x + x ln x x δ+3 ln 2 x e x / ln x I x > e, C where 0 < δ <,henex =0andT = I > e, h =, E X 2+δ <. Bu δ e / ln e / ln h= is no a slowly varying funcion a infiniy. e / ln In fac, we have he following resul. Theorem 2.3 Suose X is a real random variable and δ >0.Then E X 2+δ < if and only if δ T 0 and s δ Ts ds 0 as. Remark 2.5 If δ Tisboundedas for some δ >0,henwehaveE X 2+α < for every α 0, δfromtheorem2.3. Remark 2.6 Le X be a random variable wih zero mean. If here exis osiive consans C and C 2 such ha C l δ T C 2 l forsufficienlylarge and some δ >0,where l is a slowly varying funcion a infiniy, hen from Lemma 2.24 and Theorem 2.3,we have E X 2+δ < ls s ds 0 as. 3 Proofsofhemainresuls Proof of Theorem 2. Wihou loss of generaliy, we suose ha σ 2 =,0<ɛ <.Since P S n n / ɛ =2 n 2 /2 ɛ + R n, 3. where R n = P S n n / ɛ n / /2 ɛ + n / /2 ɛ P S n n / ɛ. From 3., we have ɛ 2r /2 λ r, ɛ r E N 2r /2 =2ɛ 2r /2 n r/ 2 n 2 /2 ɛ r E N 2r /2 + ɛ 2r /2 n r/ 2 R n. 3.2

5 HeJournal of Inequaliies and Alicaions 203, 203:378 Page 5 of h:// By Lemma 2., in order o rove Theorem 2., we only need o esimae ɛ 2r /2 nr/ 2 R n. On accoun of a non-uniform esimae of he cenral limi heorem by Nagaev [8, for every x R, P Sn < x x n CE X n + x 3 By 3.3, R n a If r <3/2, hen CE X 3 n+ɛn 2 /2 3. ɛ 2r /2 n r/ 2 R n Cɛ 2r /2 n r/ 5/2 = O ɛ 2r / b If 3/2 < r <3,hen ɛ 2r /2 Cɛ 2r /2 n r/ 2 R n [ɛ 2/2 Cɛ 2r /2 n r/ 2 n + ɛn 2 /2 3 n r/ 2 n + ɛ 3 n=[ɛ 2/2 + n r/ 5/2 6 3/2 = O ɛ / c If r =3/2, hen ɛ 2r /2 [ɛ 2/2 n r/ 2 R n Cɛ /2 n + ɛ 3 n=[ɛ 2/2 + n 6 3/2 = O ɛ /2 log. 3.6 ɛ From 2., 3.2, 3.4, 3.5and3.6, we obain 2.2. This comlees he roof of ar. 2 By he inequaliy in Osiov and Perov [9, here exiss a bounded and decreasing funcion ψuonheinerval0, suchhalim u ψu=0and P S n < x x nσ ψ n + x n δ/2 + x. 2+δ Le x = n 2 /2 ɛ,wehave R n 2ψ n+n 2 /2 ɛ n δ/2 +n 2 /2 ɛ 2+δ,soha: a If 2 < r <+δ/2,hen ɛ 2r /2 n r/ 2 R n ɛ 2r /2 n r/ 2 δ/2 = O ɛ 2r /2. 3.7

6 HeJournal of Inequaliies and Alicaions 203, 203:378 Page 6 of h:// b If + δ/2 < r <2+δ, hen by noicing ha lim u ψu = 0 for any η >0,here exiss a naural number N 0 such ha ψ n<η whenever n > N 0.Weconcludeha ɛ 2r /2 Cɛ 2r /2 n r/ 2 R n 2n r/ 2 ψ n + n 2 /2 ɛ n δ/2 + ɛn 2 /2 2+δ N0 Cɛ 2r /2 n r/ 2 δ/2 ψ [ɛ 2/2 n+η n r/ 2 δ/2 + Cɛ 2r /2 2 δ ψ ɛ /2 n=n 0 + n=[ɛ 2/2 + n r/ 2 δ/2 / /22+δ ɛ 2r /2 N r/ δ/2 0 + Cηɛ δ/2 + Cψ ɛ /2 ɛ δ/2 = o ɛ δ/ c If r =+δ/2,hen ɛ 2r /2 n r/ 2 R n = O ɛ δ/2 log. 3.9 ɛ By 2. and combining wih 3.2, 3.7, 3.8and3.9, we obain 2.3, which comlees he roof of ar 2. 3 We make use of he following large deviaion esimae in Perov [0: P S n < x x nσ C, x >0. n + x q C So, R n n+ɛn. Hence we have he following. 2 /2 q a If r <3/2, hen ɛ 2r /2 n r/ 2 R n ɛ 2r /2 b If r >3/2, hen r 5 2 2q q 2 ɛ 2r /2 n r/ 2 R n Cɛ 2r /2 n r/ 5/2 = O ɛ 2r / <. By noing ha q > 2r 3 2,weobain [ɛ 2/2 Cɛ 2r /2 n r/ 2 + ɛn 2 /2 q n + Cɛ 2r /2 q n r/ 2 /2 n=[ɛ 2/2 + n r/ 2 /2 2 q/2 = O ɛ /2. 3.

7 HeJournal of Inequaliies and Alicaions 203, 203:378 Page 7 of h:// c If r =3/2, hen ɛ 2r /2 [ɛ 2/2 n r/ 2 R n Cɛ /2 n + Cɛ/2 q n=[ɛ 2/2 + n 2 q/2 = O ɛ /2 log. 3.2 ɛ By 2., from 3.2, 3.0, 3. and3.2, we have 2.4, which comlees he roof of ar 3. Proof of Theorem 2.2 We wrie ɛ 2 λ 2,/β+ ɛ 2β + 2ɛ 2 = n 2β e 2/2 d 2π ɛn β+/2 2β + [ɛ 4/2β+ + ɛ 2 + n=[ɛ 4/2β+ + n 2β P S n ɛn β+ 2 e 2/2 d 2π ɛn β+/2 =: I + I 2 + I Firs, according o Lemma 2.,wehave Oɛ 2, 2 I = < β < 2, Oɛ 4/2β+, β For I 3, alying Lemma 2.3 of Xie and He [, and leing x =2y = n β+ ɛ,weobain P S n n β+ ɛ np X 2 nβ+ ɛ +8e 2 ɛ 4 n 4β Observing he following fac 2 e 2/2 d =2 n β+ 2 ɛ 2ϕnβ+/2 ɛ = O ɛ 5 n 5β 5/2, 3.6 2π ɛn β+/2 n β+/2 ɛ from 3.5and3.6, we have I 3 ɛ 2 ɛ 2 n=[ɛ 4/2β+ + n=[ɛ 4/2β+ + + Cɛ 3 n=[ɛ 4/2β+ + n 2β P S n ɛn β+ + ɛ 2 n=[ɛ 4/2β+ + n 2β+ P X > ɛnβ+ + Cɛ 2 2 n 3β 5/2 n=[ɛ 4/2β+ + 2n 2β e 2/2 d 2π ɛn β+/2 n 2β 2

8 HeJournal of Inequaliies and Alicaions 203, 203:378 Page 8 of h:// ɛ 2 ɛ 2 ɛ 2 Cɛ 2 C n=[ɛ 4/2β+ + n=[ɛ 4/2β+ + n 2β+ n 2β+ k=[ɛ 4/2β+ + k=[ɛ 4/2β+ + k=[ɛ 4/2β+ + x 2 nβ+ ɛ k=n k n 2β+ k 2β+2 dfx+o ɛ 2 + O ɛ 3 2 kβ+ ɛ x< 2 k+β+ ɛ 2 kβ+ ɛ x< 2 k+β+ ɛ 2 kβ+ ɛ x< 2 k+β+ ɛ 2 kβ+ ɛ x< 2 k+β+ ɛ C x 2 dfx+o ɛ 2 x 2 ɛ 2β+3/2β+ = CT ɛ 2β+3/2β+ + O ɛ 2. dfx+o ɛ 2 dfx+o ɛ 2 dfx+o ɛ 2 x 2 dfx+o ɛ 2 Using he assumion on T and Lemma 2.2, we can obain Oɛ 2, 2 < β minδ, 4 2, I 3 = Oɛ /2β+, β 4, δ, 3.7 Oɛ δ/2β+ lɛ /2β+, β 2 + δ 4,0<δ <. For I 2, by Bikelis s inequaliy see [2, we have [ɛ 4/2β+ I 2 ɛ 2 [ɛ 2/2β+ ɛ 2 [ɛ 4/2β+ + ɛ Cn 2β + ɛn β+/2 3 n +ɛn β+/2 n 2β /2 n n=[ɛ 2/2β ɛn β+/2 n 0 Tv dv Tv dv +ɛn β+/2 n β 2 n Tv dv. 0 Now, he roofof Theorem 2.2 will be divided ino he following cases. Case of δ >. Noing ha T EX 2 =,leδ be a real number such ha < δ < δ, by Lemma 2.2, lim δ δ l = 0. Therefore, here is a real number T 0 >0suchha l < whenever δ δ > T 0.Then 0 T d 0 T d + T d C + d <. T 0 δ

9 HeJournal of Inequaliies and Alicaions 203, 203:378 Page 9 of h:// We have [ɛ 2/2β+ I 2 Cɛ 2 [ɛ 4/2β+ + n 2β /2 + Cɛ n=[ɛ 2/2β+ + n β 2 Oɛ 2, 2 < β 4, = Oɛ 2 log ɛ, β = 4, 3.8 Oɛ /2β+, β > 4. From 3.3, 3.4, 3.7and3.8, we obain 2.5. Case 2 of 0 < δ <. a If 2 < β < 2 + δ 4,hen n2β /2 < and 4β+ δ l d <. Making use of Lemma , we have [ɛ 2/2β+ I 2 Cɛ 2 [ɛ 4/2β+ + Cɛ [ɛ 2/2β+ Cɛ 2 n 2β /2 + n=[ɛ 2/2β+ + 2 n n β 2 + T d 2ɛn β+ n 2β /2 n δ l n+o ɛ 2 [ɛ 4/2β+ + Cɛ /2β+ + Cɛ n=[ɛ 2/2β+ + ɛ 2/2β+ Cɛ 2 x 2β /2 x δ l x dx + Cɛ δ T d n β 2 2n β+ ɛ δ l 2n β+ ɛ ɛ 2/2β+ x β 2 l 2x β+ ɛ x β+ δ dx + O ɛ 2 ɛ /2β+ Cɛ 2 4β+ δ l d + C Cɛ 2 + Cɛ δ/2β+ l ɛ /2β+ ɛ /2β+ l +δ d + O ɛ 2 = O ɛ b If β 2 + δ 4,henwehave I 2 Cɛ 2 ɛ 4β+/2β+ + 2ɛ /2β+ T d + Cɛ /2β+ + Cɛ δ/2β+ l ɛ /2β+ Cɛ /2β+ + 2ɛ δ/2β+ l ɛ /2β+ + Cɛ δ/2β+ l ɛ /2β+ Cɛ δ/2β+ l ɛ /2β

10 HeJournal of Inequaliies and Alicaions 203, 203:378 Page 0 of h:// Therefore Oɛ 2, 2 I 2 = < β 2 + δ 4, Oɛ δ/2β+ lɛ /2β+, β 2 + δ Combining he esimae wih 3.and3.4, by 3.0, his imlies ha 2.6 follows. Case 3 of δ =. a If 2 < β < 4,hen n2β 2 <.Wehave I 2 Cɛ 2 + ɛ /2β+ [ɛ 2/2β+ T d n 2β /2 ɛ + Cɛ + 2β+3 2β+ [ɛ 4/2β+ T d n β 2 n=[ɛ 2/2β+ + /2β+ Cɛ + 2 l ɛ /2β+ Cɛ + 2 l d b If β > 4,henwehave ɛ I 2 Cɛ 2 ɛ 2β+ + 2 /2β+ 2β+/2 l /2β+ Cɛ + /2β+ l ɛ 2β+3/2β+ Cɛ + /2β+ l d c If β = 4,henwehave I 2 Cɛ 2 log ɛ Cɛ 2 log ɛ ɛ 2 l + + ɛ 5 l 2β+3 ɛ 2β+ d + ɛ /2β+ + l d ɛ 2β+3/2β+ d + Cɛ /2β+ l + d + ɛ /2β+ + ɛ 2β+3/2β+ l d d ɛ 5 d + Cɛ 2 l + d d so ha Oɛ 2 + ɛ /2β+ I 2 = Oɛ 2 + ɛ 5 Oɛ /2β+ + ɛ 2β+3/2β+ l d, 2 < β 4, l d log ɛ, β = 4, l d, β > Combining he esimae wih 3.4and3.7, by 3.3, his imlies ha 2.7 follows, and hence Theorem 2.2is roved.

11 HeJournal of Inequaliies and Alicaions 203, 203:378 Page of h:// Proof of Theorem 2.3 Se T =E X 2+δ I X >. Firs, noe ha E X 2+δ I X > = x 2+δ dfx x > = = x > = δ x 2 x δy δ δy δ dy dfx+ δ x >y s δ Ts ds + δ T. x 2 dfx dy + δ T x 2 dfx x > We have T =δ s δ Ts ds + δ T. Since s δ Ts ds 0, δ T 0, we have T 0 δ T 0 and Nex, i is easy o ge s δ Ts ds 0 as. E X 2+δ < T 0 as. From he above facs, he roof of Theorem 2.3 is comlee. Comeing ineress The auhor declares ha hey have no comeing ineress. Acknowledgemens The auhor would like o hank he referee for helful commens. Received: 3 February 203 Acceed: 8 July 203 Published: Augus 203 References. Heyde, CC: A sulemen o he srong law of large numbers. J. Al. Probab. 2, Klesov, OI: On he convergence rae in a heorem of Heyde. Theory Probab. Mah. Sa. 49, He, JJ, Xie, TF: Asymoic roery for some series of robabiliy. Aca Mah. Al. Sin. 29, Gu, A, Seinebach, J: Convergence raes in recise asymoics. J. Mah. Anal. Al. 390, Gu, A, Seinebach, J: Precise asymoics-a general aroach. Aca Mah. Hung. 38, Gu, A, Seinebach, J: Correcion o Convergence raes in recise asymoics. Prerin h://www2.mah.uu.se/~allan/90correcion.df Bingham, NH, Goldie, CM, Teugels, JL: Regular Variaion. Cambridge Universiy Press, Cambridge Nagaev, SV: Some limi heorems for large deviaion. Theory Probab. Al. 0, Osiov, LV, Perov, VV, On an esimae of remainder erm in he cenral limi heorem. Theory Probab. Al. 2, Perov, VV: Limi Theorems of Probabiliy Theory. Oxford Universiy Press, Oxford 995. Xie, TF, He, JJ: Rae of convergence in a heorem of Heyde. Sa. Probab. Le. 82, Bikelis, A: Esimaes of he remainder in he cenral limi heorem. Lie. Ma. Rink. 6, doi:0.86/ x Cie his aricle as: He: A noe o he convergence raes in recise asymoics. Journal of Inequaliies and Alicaions :378.