We obtain an ahnost sure central limit theorem(ASCLT)for heavily trimmed sums.We also prove a function-typed ASCLT under the same conditions that assure measurable functions to satisfy the ASCLT for the partial sums o...We obtain an ahnost sure central limit theorem(ASCLT)for heavily trimmed sums.We also prove a function-typed ASCLT under the same conditions that assure measurable functions to satisfy the ASCLT for the partial sums of i.i.d,random variables with EX_1=0,EX_1~2=1.展开更多
Let{X n}be a sequence of random variables and X n1X n2…X nn their order statistics.In this paper a central limit theorem and a strong law of large numbers for randomly trimmed sums T n=βn i=αn+1 X ni are establishe...Let{X n}be a sequence of random variables and X n1X n2…X nn their order statistics.In this paper a central limit theorem and a strong law of large numbers for randomly trimmed sums T n=βn i=αn+1 X ni are established in the case thatαn andβn are positive integer-valued random variables such thatαn/n andβn/n converge to random variablesαandβrespectively with 0α<β1 in certain sense,and{X n}is aφ-mixing sequence.展开更多
Let θ∈^d be a unit vector and let X, X1, X2,…… be a sequence of i.i.d. Xd-valued random vectors attracted to operator semi-stable laws. For each integer n ≥1, let X1,≤……≤ Xn,n denote the order statistics of X...Let θ∈^d be a unit vector and let X, X1, X2,…… be a sequence of i.i.d. Xd-valued random vectors attracted to operator semi-stable laws. For each integer n ≥1, let X1,≤……≤ Xn,n denote the order statistics of X1, X2,..., Xn according to priority of index, namely ｜（X1,nθ）｜≥…≥ [（Xn,n,θ）1, where （., .） is an inner product on Rd. For all integers r ≥ 0, define by （r）Sn =∑n-r i=1Xi,n the trimmed sum. In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums （r）Sn. Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded. A stochastically compactness of （r）Sn is obtained.展开更多
The authors first derive the normal expansion of the joint density function of two orderstatistics from the uniform distribution and then, using the approximation, establish a wayto estimate the normal convergence rat...The authors first derive the normal expansion of the joint density function of two orderstatistics from the uniform distribution and then, using the approximation, establish a wayto estimate the normal convergence rate for trimmed sums. For applications, the convergencerates for the intermediately trimmed sums and heavily trimmed surns are found out.展开更多
Let {X, X_; ∈N^d} be a field of i.i.d, random variables indexed by d-tuples of positive integers and taking values in a Banach space B and let X_^((r))=X_(m) if ‖X_‖ is the r-th maximum of {‖X_‖; ≤. Let S_=∑(≤...Let {X, X_; ∈N^d} be a field of i.i.d, random variables indexed by d-tuples of positive integers and taking values in a Banach space B and let X_^((r))=X_(m) if ‖X_‖ is the r-th maximum of {‖X_‖; ≤. Let S_=∑(≤)X_ and ^((r))S_=S_-(X_^((1))+…+X_^((r)). We approximate the trimmed sums ^((r))_n, by a Brownian sheet and obtain sufficient and necessary conditions for ^((r))S_ to satisfy the compact and functional laws of the iterated logarithm. These results improve the previous works by Morrow (1981), Li and Wu (1989) and Ledoux and Talagrand (1990).展开更多
基金Supported by the National Natural Science Foundation of China(No.10071003)Beijing Municipal Education Commission(KM200310028107)
文摘We obtain an ahnost sure central limit theorem(ASCLT)for heavily trimmed sums.We also prove a function-typed ASCLT under the same conditions that assure measurable functions to satisfy the ASCLT for the partial sums of i.i.d,random variables with EX_1=0,EX_1~2=1.
文摘Let{X n}be a sequence of random variables and X n1X n2…X nn their order statistics.In this paper a central limit theorem and a strong law of large numbers for randomly trimmed sums T n=βn i=αn+1 X ni are established in the case thatαn andβn are positive integer-valued random variables such thatαn/n andβn/n converge to random variablesαandβrespectively with 0α<β1 in certain sense,and{X n}is aφ-mixing sequence.
基金Supported by National Natural Science Foundation of China(Grant No.11071076)NSF of Zhejiang Province(Grant No.LY14A010025)
文摘Let θ∈^d be a unit vector and let X, X1, X2,…… be a sequence of i.i.d. Xd-valued random vectors attracted to operator semi-stable laws. For each integer n ≥1, let X1,≤……≤ Xn,n denote the order statistics of X1, X2,..., Xn according to priority of index, namely ｜（X1,nθ）｜≥…≥ [（Xn,n,θ）1, where （., .） is an inner product on Rd. For all integers r ≥ 0, define by （r）Sn =∑n-r i=1Xi,n the trimmed sum. In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums （r）Sn. Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded. A stochastically compactness of （r）Sn is obtained.
文摘The authors first derive the normal expansion of the joint density function of two orderstatistics from the uniform distribution and then, using the approximation, establish a wayto estimate the normal convergence rate for trimmed sums. For applications, the convergencerates for the intermediately trimmed sums and heavily trimmed surns are found out.
基金Supported by National Natural Science Foundation of China (No. 10071072)
文摘Let {X, X_; ∈N^d} be a field of i.i.d, random variables indexed by d-tuples of positive integers and taking values in a Banach space B and let X_^((r))=X_(m) if ‖X_‖ is the r-th maximum of {‖X_‖; ≤. Let S_=∑(≤)X_ and ^((r))S_=S_-(X_^((1))+…+X_^((r)). We approximate the trimmed sums ^((r))_n, by a Brownian sheet and obtain sufficient and necessary conditions for ^((r))S_ to satisfy the compact and functional laws of the iterated logarithm. These results improve the previous works by Morrow (1981), Li and Wu (1989) and Ledoux and Talagrand (1990).
