Consider a sequence of i.i.d.positive random variables.An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit theorem holds under a...Consider a sequence of i.i.d.positive random variables.An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit theorem holds under a fairly general condition on the weight dk= k-1 exp(lnβk),0≤β〈1.And in a sense,our results have reached the optimal form.展开更多
Let {X n,n≥1} be a stationary LNQD or NA sequence satisfying EX 1=μ,EX 2 1<∞ and (Var S n)/n→σ 2 as n→∞.In this paper a class of self-normalized central limit theorems and estimators of Var S n are ...Let {X n,n≥1} be a stationary LNQD or NA sequence satisfying EX 1=μ,EX 2 1<∞ and (Var S n)/n→σ 2 as n→∞.In this paper a class of self-normalized central limit theorems and estimators of Var S n are studied.The weak and strong consistency of the estimators of Var S n are presented.展开更多
In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the p...In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.展开更多
In this paper,we discuss the suffcient condition for complete convergence of partial sums of squares of the ARCH sequence,at the same time a moment inequality of maximum partial sums for squares of the ARCH sequence i...In this paper,we discuss the suffcient condition for complete convergence of partial sums of squares of the ARCH sequence,at the same time a moment inequality of maximum partial sums for squares of the ARCH sequence is proved.展开更多
Let be a strictly stationary sequence of ρ?-mixing random variables. We proved the almost sure central limit theorem, containing the general weight sequences, for the partial sums , where , . The result generalizes a...Let be a strictly stationary sequence of ρ?-mixing random variables. We proved the almost sure central limit theorem, containing the general weight sequences, for the partial sums , where , . The result generalizes and improves the previous results.展开更多
Let {Xni, 1 ≤ n,i 〈 ∞} be an an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 〈 an ↑∞ . The limiting behavior of maximum partial sums 1/an max 1≤k≤n|^k∑i=1 Xni| is inv...Let {Xni, 1 ≤ n,i 〈 ∞} be an an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 〈 an ↑∞ . The limiting behavior of maximum partial sums 1/an max 1≤k≤n|^k∑i=1 Xni| is investigated and some new results are obtained. The results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2].展开更多
Let {Xn, n≥1} be a martingale difference sequence and {a nk , 1?k?n,n?1} an array of constant real numbers. The limiting behavior of weighted partial sums ∑ k=1 n a nk X k is investigated and some new results are ob...Let {Xn, n≥1} be a martingale difference sequence and {a nk , 1?k?n,n?1} an array of constant real numbers. The limiting behavior of weighted partial sums ∑ k=1 n a nk X k is investigated and some new results are obtained.展开更多
Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 |Sn/n^1/r|^p(0 〈 r...Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 |Sn/n^1/r|^p(0 〈 r 〈 2,p 〉 0) are given,which are the same as that in the independent case.展开更多
Using suitable self-normalization for partial sums of i.i.d.random variables,Griffin and Kuelbs established the law of the iterated logarithm for all distributions in the domain of attraction of a normal law.We obtain...Using suitable self-normalization for partial sums of i.i.d.random variables,Griffin and Kuelbs established the law of the iterated logarithm for all distributions in the domain of attraction of a normal law.We obtain the corresponding results for Studentized increments of partial sums under thesame condition.展开更多
In this paper, we study the joint limit distributions of point processes of exceedances and partial sums of multivariate Gaussian sequences and show that the point processes and partial sums are asymptotically indepen...In this paper, we study the joint limit distributions of point processes of exceedances and partial sums of multivariate Gaussian sequences and show that the point processes and partial sums are asymptotically independent under some mild conditions. As a result, for a sequence of standardized stationary Gaussian vectors, we obtain that the point process of exceedances formed by the sequence (centered at the sample mean) converges in distribution to a Poisson process and it is asymptotically independent of the partial sums. The asymptotic joint limit distributions of order statistics and partial sums are also investigated under different conditions.展开更多
Let {Xi}i=1^∞ be a standardized stationary Gaussian sequence with covariance function τ(n) =EX1Xn+1, Sn =∑i=1^nXi,and X^-n=Sn/n.And let Nn be the point process formed by the exceedances of random level (x/√2 l...Let {Xi}i=1^∞ be a standardized stationary Gaussian sequence with covariance function τ(n) =EX1Xn+1, Sn =∑i=1^nXi,and X^-n=Sn/n.And let Nn be the point process formed by the exceedances of random level (x/√2 log n+√2 log n-log(4π log n)/2√log n) √1-τ(n) + X^-n by X1,X2,…, Xn. Under some mild conditions, Nn and Sn are asymptotically independent, and Nn converges weakly to a Poisson process on (0,1].展开更多
We give a systematic account of results which assure positivity and boundedness of partial sums of cosine or sine series. New proofs of recent results are sketched.
In this paper, we obtain the moment conditions for the supermun of normed sums of ρ^--mixing random variables by using the Rosenthal-type inequality for Maximum partial sums of ρ^--mixing random variables. The resul...In this paper, we obtain the moment conditions for the supermun of normed sums of ρ^--mixing random variables by using the Rosenthal-type inequality for Maximum partial sums of ρ^--mixing random variables. The result obtained generalize the results of Chen(2008) and extend those to negatively associated sequences and ρ^--mixing random variables.展开更多
In [1] and some following publications, Tadmor and Gelb took up a well known property of conjugate Fourier series in 1-d, namely the property to detect jump discontinuities in given spectral data. In fact, this proper...In [1] and some following publications, Tadmor and Gelb took up a well known property of conjugate Fourier series in 1-d, namely the property to detect jump discontinuities in given spectral data. In fact, this property of conjugate series is known for quite a long time. The research in papers around the year 1910 shows that there were also other means of detecting jumps observed and analysed. We review the classical results as well as the results of Gelb and Tadmor and demonstrate their discrete case using different estimates in all detail. It is worth noting that the techniques presented are not global but local techniques. Edges are a local phenomenon and can only be found appropriately by local means. Furthermore, applying a different approach in the proof of the main estimate leads to weaker preconditions in the discrete case. Finally an outlook to a two-dimensional approach based on the work of Móricz, in which jumps in the mixed second derivative of a 2-d function are detected, is made.展开更多
This paper concerns with the point-wise convergence of a wavelet series for L-p functions and the characterization of Holder class by both the partial sums approximation and the best approximation.
This paper deals with the order of magnitude of the partial sums of the spherical harmonic series and its convergence rate in Bessel potential spaces. The partial results obtained in the paper are the analogue of tho...This paper deals with the order of magnitude of the partial sums of the spherical harmonic series and its convergence rate in Bessel potential spaces. The partial results obtained in the paper are the analogue of those on the circle.展开更多
The rate of uniform strong approzimation of Marcinkiewicz type for multivariable continuous functions is obtained in this paper as follows:‖1/k+1 ^k∑j=0 |Sj(f)-f|^q‖≤C/k+1^k∑j=0 Ej^q(f),where Sj(f) den...The rate of uniform strong approzimation of Marcinkiewicz type for multivariable continuous functions is obtained in this paper as follows:‖1/k+1 ^k∑j=0 |Sj(f)-f|^q‖≤C/k+1^k∑j=0 Ej^q(f),where Sj(f) denotes the square partial Fourier sum off and Ej(f) denotes the square best approximation of f by trigonometric polynomials of degree(j,j,…,j),j=0,1,2.…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
Let {Xn,n 1) be a sequence of mixing random variables, S(n)S, >0, d=1 or-1.This note investigates that where is an arbitrary but fixed positive number. As application we give some results on the convergence rates...Let {Xn,n 1) be a sequence of mixing random variables, S(n)S, >0, d=1 or-1.This note investigates that where is an arbitrary but fixed positive number. As application we give some results on the convergence rates for randomly indexed partial sums. Some known results are extended and generalized.展开更多
For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of ...For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which (i) lim sup n→∞ ||Sn||/an〈∞ a.s.and ∞ ∑n=1(1/n)P(||Sn||/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞) are equivalent;(ii) For all constants λ ∈ [0, ∞),lim sup ||Sn||/an =λ a.s.and ^∞∑ n=1(1/n) P(||Sn||/an ≥ε){〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.展开更多
基金Supported by the National Natural Science Foundation of China(11061012)Project Supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning([2011]47)the Guangxi Natural Science Foundation of China(2012GXNSFAA053010)
文摘Consider a sequence of i.i.d.positive random variables.An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit theorem holds under a fairly general condition on the weight dk= k-1 exp(lnβk),0≤β〈1.And in a sense,our results have reached the optimal form.
基金the National Natural Science Foundation of China(1 0 0 71 0 72 )
文摘Let {X n,n≥1} be a stationary LNQD or NA sequence satisfying EX 1=μ,EX 2 1<∞ and (Var S n)/n→σ 2 as n→∞.In this paper a class of self-normalized central limit theorems and estimators of Var S n are studied.The weak and strong consistency of the estimators of Var S n are presented.
文摘In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.
基金Supported by Natural Science Foundation of Shanxi Province(2007011014) Supported by Science Foundation of Shanxi Datong Unversity(2007K06) Supported by 11-5 Program Fund of Shanxi Province(GH-09090)
文摘In this paper,we discuss the suffcient condition for complete convergence of partial sums of squares of the ARCH sequence,at the same time a moment inequality of maximum partial sums for squares of the ARCH sequence is proved.
基金supported by National Natural Science Foundation of China(11361019).
文摘Let be a strictly stationary sequence of ρ?-mixing random variables. We proved the almost sure central limit theorem, containing the general weight sequences, for the partial sums , where , . The result generalizes and improves the previous results.
文摘Let {Xni, 1 ≤ n,i 〈 ∞} be an an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 〈 an ↑∞ . The limiting behavior of maximum partial sums 1/an max 1≤k≤n|^k∑i=1 Xni| is investigated and some new results are obtained. The results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2].
基金SupportedbytheNationalNaturalScienceFoundationofChina (No .10 0 710 5 8)and (No .10 0 710 19)
文摘Let {Xn, n≥1} be a martingale difference sequence and {a nk , 1?k?n,n?1} an array of constant real numbers. The limiting behavior of weighted partial sums ∑ k=1 n a nk X k is investigated and some new results are obtained.
基金Supported by the National Natural Science Foundation of China (60874004)
文摘Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 |Sn/n^1/r|^p(0 〈 r 〈 2,p 〉 0) are given,which are the same as that in the independent case.
基金Project supported by the National Natural Science Foundation of Chinaan NSERC Canada grant of M.Csorgo at Carletoa University of Canada+1 种基金the Fok Yingtung Education Foundationan NSERC Canada Scientific Exchange Award at Carleton University
文摘Using suitable self-normalization for partial sums of i.i.d.random variables,Griffin and Kuelbs established the law of the iterated logarithm for all distributions in the domain of attraction of a normal law.We obtain the corresponding results for Studentized increments of partial sums under thesame condition.
基金Supported by National Natural Science Foundation of China(Grant No.11171275)the Program for Excellent Talents in Chongqing Higher Education Institutions(Grant No.120060-20600204)supported by the Swiss National Science Foundation Project(Grant No.200021-134785)
文摘In this paper, we study the joint limit distributions of point processes of exceedances and partial sums of multivariate Gaussian sequences and show that the point processes and partial sums are asymptotically independent under some mild conditions. As a result, for a sequence of standardized stationary Gaussian vectors, we obtain that the point process of exceedances formed by the sequence (centered at the sample mean) converges in distribution to a Poisson process and it is asymptotically independent of the partial sums. The asymptotic joint limit distributions of order statistics and partial sums are also investigated under different conditions.
基金Supported by the Program for Excellent Talents in Chongqing Higher Education Institutions (120060-20600204)
文摘Let {Xi}i=1^∞ be a standardized stationary Gaussian sequence with covariance function τ(n) =EX1Xn+1, Sn =∑i=1^nXi,and X^-n=Sn/n.And let Nn be the point process formed by the exceedances of random level (x/√2 log n+√2 log n-log(4π log n)/2√log n) √1-τ(n) + X^-n by X1,X2,…, Xn. Under some mild conditions, Nn and Sn are asymptotically independent, and Nn converges weakly to a Poisson process on (0,1].
文摘We give a systematic account of results which assure positivity and boundedness of partial sums of cosine or sine series. New proofs of recent results are sketched.
基金Supported by the National Science Foundation of China(10661006)Supported by Innovation Project of Guangxi Graduate Education(2007105960812M18)
文摘In this paper, we obtain the moment conditions for the supermun of normed sums of ρ^--mixing random variables by using the Rosenthal-type inequality for Maximum partial sums of ρ^--mixing random variables. The result obtained generalize the results of Chen(2008) and extend those to negatively associated sequences and ρ^--mixing random variables.
文摘In [1] and some following publications, Tadmor and Gelb took up a well known property of conjugate Fourier series in 1-d, namely the property to detect jump discontinuities in given spectral data. In fact, this property of conjugate series is known for quite a long time. The research in papers around the year 1910 shows that there were also other means of detecting jumps observed and analysed. We review the classical results as well as the results of Gelb and Tadmor and demonstrate their discrete case using different estimates in all detail. It is worth noting that the techniques presented are not global but local techniques. Edges are a local phenomenon and can only be found appropriately by local means. Furthermore, applying a different approach in the proof of the main estimate leads to weaker preconditions in the discrete case. Finally an outlook to a two-dimensional approach based on the work of Móricz, in which jumps in the mixed second derivative of a 2-d function are detected, is made.
文摘This paper concerns with the point-wise convergence of a wavelet series for L-p functions and the characterization of Holder class by both the partial sums approximation and the best approximation.
文摘This paper deals with the order of magnitude of the partial sums of the spherical harmonic series and its convergence rate in Bessel potential spaces. The partial results obtained in the paper are the analogue of those on the circle.
基金Supported by NSF of China, under the Grant 10471010.
文摘The rate of uniform strong approzimation of Marcinkiewicz type for multivariable continuous functions is obtained in this paper as follows:‖1/k+1 ^k∑j=0 |Sj(f)-f|^q‖≤C/k+1^k∑j=0 Ej^q(f),where Sj(f) denotes the square partial Fourier sum off and Ej(f) denotes the square best approximation of f by trigonometric polynomials of degree(j,j,…,j),j=0,1,2.…
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Let {Xn,n 1) be a sequence of mixing random variables, S(n)S, >0, d=1 or-1.This note investigates that where is an arbitrary but fixed positive number. As application we give some results on the convergence rates for randomly indexed partial sums. Some known results are extended and generalized.
基金the Natural Sciences and Engineering Research Council of Canada
文摘For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which (i) lim sup n→∞ ||Sn||/an〈∞ a.s.and ∞ ∑n=1(1/n)P(||Sn||/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞) are equivalent;(ii) For all constants λ ∈ [0, ∞),lim sup ||Sn||/an =λ a.s.and ^∞∑ n=1(1/n) P(||Sn||/an ≥ε){〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.