He Journal of Inequaliies and Alicaions 203, 203:378 h://www.journalofinequaliiesandalicaions.com/conen/203//378 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 >0. 203 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.
HeJournal of Inequaliies and Alicaions 203, 203:378 Page 2 of h://www.journalofinequaliiesandalicaions.com/conen/203//378 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
HeJournal of Inequaliies and Alicaions 203, 203:378 Page 3 of h://www.journalofinequaliiesandalicaions.com/conen/203//378 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. 2.3 3 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 4. 2.6 3 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 > 4. 2.7 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
HeJournal of Inequaliies and Alicaions 203, 203:378 Page 4 of h://www.journalofinequaliiesandalicaions.com/conen/203//378 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
HeJournal of Inequaliies and Alicaions 203, 203:378 Page 5 of h://www.journalofinequaliiesandalicaions.com/conen/203//378 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 3. 3.3 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 /2. 3.4 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 /2. 3.5 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
HeJournal of Inequaliies and Alicaions 203, 203:378 Page 6 of h://www.journalofinequaliiesandalicaions.com/conen/203//378 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/2. 3.8 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 /2. 3.0 <. 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.
HeJournal of Inequaliies and Alicaions 203, 203:378 Page 7 of h://www.journalofinequaliiesandalicaions.com/conen/203//378 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 3. 3.3 Firs, according o Lemma 2.,wehave Oɛ 2, 2 I = < β < 2, Oɛ 4/2β+, β 2. 3.4 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 2. 3.5 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
HeJournal of Inequaliies and Alicaions 203, 203:378 Page 8 of h://www.journalofinequaliiesandalicaions.com/conen/203//378 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+ + 0 +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
HeJournal of Inequaliies and Alicaions 203, 203:378 Page 9 of h://www.journalofinequaliiesandalicaions.com/conen/203//378 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 2.22-3, 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 2. 3.9 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+. 3.20
HeJournal of Inequaliies and Alicaions 203, 203:378 Page 0 of h://www.journalofinequaliiesandalicaions.com/conen/203//378 Therefore Oɛ 2, 2 I 2 = < β 2 + δ 4, Oɛ δ/2β+ lɛ /2β+, β 2 + δ 4. 3.2 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. 3.22 ɛ 2β+3/2β+ d + Cɛ /2β+ l + d + ɛ /2β+ + ɛ 2β+3/2β+ l d d. 3.23 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 > 4. 3.24 Combining he esimae wih 3.4and3.7, by 3.3, his imlies ha 2.7 follows, and hence Theorem 2.2is roved.
HeJournal of Inequaliies and Alicaions 203, 203:378 Page of h://www.journalofinequaliiesandalicaions.com/conen/203//378 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, 903-907 975 2. Klesov, OI: On he convergence rae in a heorem of Heyde. Theory Probab. Mah. Sa. 49, 83-87 994 3. He, JJ, Xie, TF: Asymoic roery for some series of robabiliy. Aca Mah. Al. Sin. 29, 79-86 203 4. Gu, A, Seinebach, J: Convergence raes in recise asymoics. J. Mah. Anal. Al. 390, -4 202 5. Gu, A, Seinebach, J: Precise asymoics-a general aroach. Aca Mah. Hung. 38, 365-385 203 6. Gu, A, Seinebach, J: Correcion o Convergence raes in recise asymoics. Prerin h://www2.mah.uu.se/~allan/90correcion.df 202 7. Bingham, NH, Goldie, CM, Teugels, JL: Regular Variaion. Cambridge Universiy Press, Cambridge 987 8. Nagaev, SV: Some limi heorems for large deviaion. Theory Probab. Al. 0, 24-235 965 9. Osiov, LV, Perov, VV, On an esimae of remainder erm in he cenral limi heorem. Theory Probab. Al. 2, 28-286 967 0. 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, 576-582 202 2. Bikelis, A: Esimaes of he remainder in he cenral limi heorem. Lie. Ma. Rink. 6, 323-346 966 doi:0.86/029-242x-203-378 Cie his aricle as: He: A noe o he convergence raes in recise asymoics. Journal of Inequaliies and Alicaions 203 203:378.