Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1...Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.展开更多
Let X=Σ_(i=1)^(n)a_(i)ξ_(i)be a Rademacher sum with Var(X)=1 and Z be a standard normal random variable.This paper concerns the upper bound of|P(X≤x)−P(Z≤x)|for any x∈R.Using the symmetric properties and R softwa...Let X=Σ_(i=1)^(n)a_(i)ξ_(i)be a Rademacher sum with Var(X)=1 and Z be a standard normal random variable.This paper concerns the upper bound of|P(X≤x)−P(Z≤x)|for any x∈R.Using the symmetric properties and R software,this paper gets the following improved Berry-Esseen type bound under some conditions,|P(X≤x)−P(Z≤x)|≤P(Z∈(0,a1)),∀x∈R,which is one of the modified conjecture proposed by Nathan K.and Ohad K.展开更多
设{X,Xn,n≥1}是独立同分布正态随机变量序列,EX=0且EX2=σ2>0,Sn=sum (Xk) form k=1 to n,λ(ε) =sum form (P(|Sn|≥ nε)) form n=1 to ∞.在本文中,我们证明了存在正常数C1和C2,使得对足够小的ε>0,成立下列不等式C1ε3 ≤ε...设{X,Xn,n≥1}是独立同分布正态随机变量序列,EX=0且EX2=σ2>0,Sn=sum (Xk) form k=1 to n,λ(ε) =sum form (P(|Sn|≥ nε)) form n=1 to ∞.在本文中,我们证明了存在正常数C1和C2,使得对足够小的ε>0,成立下列不等式C1ε3 ≤ε2λ(ε)-σ2+ε2 /2 ≤ C2ε3.展开更多
基金Research supported by National Nature Science Foundation of China:10471126
文摘Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.
基金supported by the National Natural Science Foundation of China(Grant No.11861029)the Hainan Provincial Natural Science Foundation of China(Grants Nos.122MS056,124MS056).
文摘Let X=Σ_(i=1)^(n)a_(i)ξ_(i)be a Rademacher sum with Var(X)=1 and Z be a standard normal random variable.This paper concerns the upper bound of|P(X≤x)−P(Z≤x)|for any x∈R.Using the symmetric properties and R software,this paper gets the following improved Berry-Esseen type bound under some conditions,|P(X≤x)−P(Z≤x)|≤P(Z∈(0,a1)),∀x∈R,which is one of the modified conjecture proposed by Nathan K.and Ohad K.
文摘设{X,Xn,n≥1}是独立同分布正态随机变量序列,EX=0且EX2=σ2>0,Sn=sum (Xk) form k=1 to n,λ(ε) =sum form (P(|Sn|≥ nε)) form n=1 to ∞.在本文中,我们证明了存在正常数C1和C2,使得对足够小的ε>0,成立下列不等式C1ε3 ≤ε2λ(ε)-σ2+ε2 /2 ≤ C2ε3.
