在希尔伯特空间的一般重对数律的精确速率  

Precise Rates in the Generalized Law of the Iterated Logarithm in the Hilbert Space

作　　者：徐明周 程琨 XU Mingzhou;CHENG Kun(School of Information Engineering,Jingdezhen Ceramic University,Jingdezhen,333403,China)

机构地区：[1]景德镇陶瓷大学信息工程学院,景德镇333403

出　　处：《应用概率统计》2022年第6期807-824,共18页Chinese Journal of Applied Probability and Statistics

基　　金：the Doctoral Scientific Research Starting Foundation of Jingdezhen Ceramic University(Grant No.102/01003002031);the Scientific Program of Department of Education of Jiangxi Province of China(Grant No.GJJ190732)。

摘　　要：设{X,X_(n),n>1}是取值于一般实可分希尔伯特空间(H,‖·‖)的具有协方差算子的独立同分布随机变量列,记S_(n)=X_(1)+X_(2)+···+X_(n),n>1.对任意m>0和a_(n)=O((ln ln n)^(−2m)),我们得到了P{‖S_(n)‖>(ϵ+a_(n))σ√n(ln ln n)^(m)}的一类加权无穷序列的重对数律的精确速率.设β_(n)(ϵ)=o(√1/ln ln n).我们也得到了对任意r>1和α>−d/2,limϵ↘√r−1[ϵ^(2)−(r−1)]^(α+d)/2∞Σn=11/n(ln n)^(r−2)(ln ln n)^(α)P{‖S_(n)‖>σψ(n)[ϵ+β_(n)(ϵ)]}=Г^(−1)(d/2)K(Σ)(r−1)^((d−2)/2)Г(α+d/2)成立,若EX=0,E[‖X‖^(2)(ln‖X‖)^(r−1)]<∞.Let{X,X_(n),n>1}be a sequence of i.i.d.random variables taking values in a real separable Hilbert space(H,‖·‖)with covariance operator,set Sn=X_(1)+X_(2)+···+X_(n),n>1.For every m>0 and an=O((ln ln n)^(−2m)),we study the precise rates in the generalized law of the iterated logarithm for a kind of weighted in nite series of P{‖S_(n)‖>(ϵ+an)σ√n(ln ln n)m}.Letβ_(n)(ϵ)=o(√1/ln ln n).We also prove that,for any r>1 andα>−d/2,limϵ↘√r−1[ϵ^(2)−(r−1)]^(α+d)/2∞Σn=11/n(ln n)^(r−2)(ln ln n)^(α)P{‖S_(n)‖>σψ(n)[ϵ+β_(n)(ϵ)]}=Г^(−1)(d/2)K(Σ)(r−1)^((d−2)/2)Г(α+d/2)holds if EX=0,E[‖X‖2(ln‖X‖)^(r−1)]<∞.

关 键 词：完全收敛 独立同分布随机变量和的尾概率 精确速率 重对数律 强估计 

分 类 号：O211.4[理学—概率论与数理统计]