For a blockwise martingale difference sequence of random elements {Vn, n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers...For a blockwise martingale difference sequence of random elements {Vn, n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form limn→∞ Vi/gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1〈 p ≤2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.展开更多
We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann's strong law of large numbers and Teicher's strong law of large nnum...We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann's strong law of large numbers and Teicher's strong law of large nnumbers for independent random variables are generalized to the case of φ -minxing random variables.展开更多
This paper studies the strong law of large numbers and the Shannom-McMillan theorem for Markov chains field on Cayley tree. The authors first prove the strong law of large number on the frequencies of states and order...This paper studies the strong law of large numbers and the Shannom-McMillan theorem for Markov chains field on Cayley tree. The authors first prove the strong law of large number on the frequencies of states and orderd couples of states for Markov chains field on Cayley tree. Then they prove the Shannon-McMillan theorem with a.e. convergence for Markov chains field on Cayley tree. In the proof, a new technique in the study the strong limit theorem in probability theory is applied.展开更多
Some strong laws of large numbers for the frequencies of occurrence of states and ordered couples of states for nonsymmetric Markov chain fields (NSMC) on Cayley trees are studied. In the proof, a new technique for ...Some strong laws of large numbers for the frequencies of occurrence of states and ordered couples of states for nonsymmetric Markov chain fields (NSMC) on Cayley trees are studied. In the proof, a new technique for the study of strong limit theorems of Markov chains is extended to the case of Markov chain fields, The asymptotic equipartition properties with almost everywhere (a,e.) convergence for NSMC on Cayley trees are obtained,展开更多
In this article, the strong laws of large numbers for array of rowwise asymptotically almost negatively associated(AANA) random variables are studied. Some sufficient conditions for strong laws of large numbers for ar...In this article, the strong laws of large numbers for array of rowwise asymptotically almost negatively associated(AANA) random variables are studied. Some sufficient conditions for strong laws of large numbers for array of rowwise AANA random variables are presented without assumption of identical distribution. Our results extend the corresponding ones for independent random variables to case of AANA random variables.展开更多
This paper introduces the concept of BC sequences and investigates some conditions which imply the strong law of large numbers for these sequences. The authors also study the strong law of large numbers for general ra...This paper introduces the concept of BC sequences and investigates some conditions which imply the strong law of large numbers for these sequences. The authors also study the strong law of large numbers for general random variable sequences. As applications of the result the authors characterize p-smoothableness of Banach space. Some generalizations of Petrov theorem, the Marcinkiewicz-Zygmund theorem and Hoffmann-J(?)rgensen and Pisier theorem are obtained.展开更多
This paper investigates some conditions which imply the strong laws of large numbers for Banach space valued random variable sequences. Some generalizations of the Marcinkiewicz-Zygmund theorem and the Hoffmann-J?rgen...This paper investigates some conditions which imply the strong laws of large numbers for Banach space valued random variable sequences. Some generalizations of the Marcinkiewicz-Zygmund theorem and the Hoffmann-J?rgensen and Pisier theorem are obtained. Key words strong law of large numbers - Banach space valued random variable sequence - p-smoothable Banach space CLC number O 211.4 - O 211.6 Foundation item: Supported by the National Natural Science Foundation of China (10071058)Biography: Gan Shi-xin (1939-), male, Professor, research direction: martingale theory, probability limiting theory and Banach space geometry theory.展开更多
In the paper,we get the precise results of Hájek-Rényi type inequalities for the partial sums of negatively orthant dependent sequences,which improve the results of Theorem 3.1and Corollary 3.2 in Kim(2006)a...In the paper,we get the precise results of Hájek-Rényi type inequalities for the partial sums of negatively orthant dependent sequences,which improve the results of Theorem 3.1and Corollary 3.2 in Kim(2006)and the strong law of large numbers and strong growth rate for negatively orthant dependent sequences.展开更多
By using a Rosenthal type inequality established in this paper, the complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are discussed for *-mixing...By using a Rosenthal type inequality established in this paper, the complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are discussed for *-mixing random fields.展开更多
In this paper, by the three series theorem of m-negatively associated(m-NA,in short) random variables and the truncation method of random variables, we mainly investigated the strong convergence properties for partial...In this paper, by the three series theorem of m-negatively associated(m-NA,in short) random variables and the truncation method of random variables, we mainly investigated the strong convergence properties for partial sums of m-NA random variables.In addition, the Khintchine-Kolmogorov convergence theorem and Kolmogorov-type strong law of large numbers for m-NA random variables are also obtained. The results obtained in the paper generalize some corresponding ones for independent random variables and some dependent random variables.展开更多
M-negatively associated random variables, which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its particular case, are introduced and studied. Large devi...M-negatively associated random variables, which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its particular case, are introduced and studied. Large deviation principles and moderate deviation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.展开更多
This paper mainly studies the strong convergence properties for weighted sums of extended negatively dependent(END,for short)random variables.Some sufficient conditions to prove the strong law of large numbers for wei...This paper mainly studies the strong convergence properties for weighted sums of extended negatively dependent(END,for short)random variables.Some sufficient conditions to prove the strong law of large numbers for weighted sums of END random variables are provided.In particular,the authors obtain the weighted version of Kolmogorov type strong law of large numbers for END random variables as a product.The results that the authors obtained generalize the corresponding ones for independent random variables and some dependent random variables.As an application,the authors investigate the errors-in-variables(EV,for short)regression models and establish the strong consistency for the least square estimators.Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analysed for illustration.展开更多
Let{Xn^-,n^-∈N^d}be a field of Banach space valued random variables, 0 〈r〈p≤2 and{an^-,k^-, (n^-,k^-) ∈ N^d × N^d ,k^-≤n^-} a triangular array of real numhers, where N^d is the d-dimensional lattice (d...Let{Xn^-,n^-∈N^d}be a field of Banach space valued random variables, 0 〈r〈p≤2 and{an^-,k^-, (n^-,k^-) ∈ N^d × N^d ,k^-≤n^-} a triangular array of real numhers, where N^d is the d-dimensional lattice (d≥1 ). Under the minimal condition that {||Xn^-|| r,n^- ∈N^d} is {|an^-,k^-|^r,(n^-,k^-)} ∈ N^d ×N^d,k^-≤n^-}-uniformly integrable, we show that ∑(k^-≤n^-)an^-,k^-,Xk^-^(L^r(or a,s,)→0 as |n^-|→∞ In the above, if 0〈r〈1, the random variables are not needed to be independent. If 1≤r〈p≤2, and Banach space valued random variables are independent with mean zero we assume the Banaeh space is of type p. If 1≤r≤p≤2 and Banach space valued random variables are not independent we assume the Banach space is p-smoothable.展开更多
In this paper, we will present some strong convergence results for sequences of ψ-mixing random variables. The results for sequences of ψ-mixing random variables generalize the corresponding results for independent ...In this paper, we will present some strong convergence results for sequences of ψ-mixing random variables. The results for sequences of ψ-mixing random variables generalize the corresponding results for independent random variable sequences without any extra conditions.展开更多
In this paper,we study the strong law of large numbers for the frequencies of occurrence of states and ordered couples of states for nonsymmetric Markov chain(NSMC) indexed by Cayley tree with any finite states.The ...In this paper,we study the strong law of large numbers for the frequencies of occurrence of states and ordered couples of states for nonsymmetric Markov chain(NSMC) indexed by Cayley tree with any finite states.The asymptotic equipartition properties with almost everywhere(a.e.) convergence for NSMC indexed by Cayley tree are obtained.This article generalizes a recent result.展开更多
We mainly study the almost sure limiting behavior of weighted sums of the form ∑ni=1 aiXi/bn , where {Xn, n ≥ 1} is an arbitrary Banach space valued random element sequence or Banach space valued martingale differen...We mainly study the almost sure limiting behavior of weighted sums of the form ∑ni=1 aiXi/bn , where {Xn, n ≥ 1} is an arbitrary Banach space valued random element sequence or Banach space valued martingale difference sequence and {an, n ≥ 1} and {bn,n ≥ 1} are two sequences of positive constants. Some new strong laws of large numbers for such weighted sums are proved under mild conditions.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 (X, Xn; n≥ 1} be a sequence of i.i.d, random variables with values in a measurable space (S,8) such that E|h(X1, X2,..., Xm)| 〈 ∞, where h is a measurable symmetric function from Sm into R = (-∞, ∞)....Let (X, Xn; n≥ 1} be a sequence of i.i.d, random variables with values in a measurable space (S,8) such that E|h(X1, X2,..., Xm)| 〈 ∞, where h is a measurable symmetric function from Sm into R = (-∞, ∞). Let {wn,i1,i2 im ; 1 ≤ i1 〈 i2 〈 …… 〈im 〈 n, n ≥ m} be a matrix array of real numbers. Motivated by a result of Choi and Sung (1987), in this note we are concerned with establishing a strong law of large numbers for weighted U-statistics with kernel h of degree m. We show that whenever SUP n≥m max1〈i1〈i2〈…〈im≤|wn i1,i2 i,im| 〈∞, where 0 = Eh(X1, X2,..., Xm). The proof of this result is based on a new general result on complete convergence, which is a fundamental tool, for array of real-valued random variables under some mild conditions.展开更多
For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ...For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p(1 ≤ p ≤ 2) Banach spaces.展开更多
We establish some strong limit theorems for a sequence of pair-wise extended lower/upper negatively dependent random variables and give some new examples of dependent random variables.
基金supported in part by the National Foundation for Science Technology Development,Vietnam (NAFOSTED) (Grant No. 101.02.32.09)
文摘For a blockwise martingale difference sequence of random elements {Vn, n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form limn→∞ Vi/gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1〈 p ≤2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.
基金Supported by the National Natural Science Foundation of China (10671149)
文摘We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann's strong law of large numbers and Teicher's strong law of large nnumbers for independent random variables are generalized to the case of φ -minxing random variables.
文摘This paper studies the strong law of large numbers and the Shannom-McMillan theorem for Markov chains field on Cayley tree. The authors first prove the strong law of large number on the frequencies of states and orderd couples of states for Markov chains field on Cayley tree. Then they prove the Shannon-McMillan theorem with a.e. convergence for Markov chains field on Cayley tree. In the proof, a new technique in the study the strong limit theorem in probability theory is applied.
基金Supported by National Basic Research Program of China(973 Program No.2007CBS14903)National Science Foundation of China(70671069)
文摘Some strong laws of large numbers for the frequencies of occurrence of states and ordered couples of states for nonsymmetric Markov chain fields (NSMC) on Cayley trees are studied. In the proof, a new technique for the study of strong limit theorems of Markov chains is extended to the case of Markov chain fields, The asymptotic equipartition properties with almost everywhere (a,e.) convergence for NSMC on Cayley trees are obtained,
基金Supported by the National Natural Science Foundation of China(lilT1001, 11201001) Supported by the Natural Science Foundation of Anhui Province(1208085QA03)+1 种基金 Supported by the Talents Youth Fund of Anhui Province Universities(2012SQRL204) Supported by th Doctoral Research Start-up Funds Projects of Anhui University(33190250)
文摘In this article, the strong laws of large numbers for array of rowwise asymptotically almost negatively associated(AANA) random variables are studied. Some sufficient conditions for strong laws of large numbers for array of rowwise AANA random variables are presented without assumption of identical distribution. Our results extend the corresponding ones for independent random variables to case of AANA random variables.
基金Supported by the National Natural Science Foundation of China(10071058)
文摘This paper introduces the concept of BC sequences and investigates some conditions which imply the strong law of large numbers for these sequences. The authors also study the strong law of large numbers for general random variable sequences. As applications of the result the authors characterize p-smoothableness of Banach space. Some generalizations of Petrov theorem, the Marcinkiewicz-Zygmund theorem and Hoffmann-J(?)rgensen and Pisier theorem are obtained.
文摘This paper investigates some conditions which imply the strong laws of large numbers for Banach space valued random variable sequences. Some generalizations of the Marcinkiewicz-Zygmund theorem and the Hoffmann-J?rgensen and Pisier theorem are obtained. Key words strong law of large numbers - Banach space valued random variable sequence - p-smoothable Banach space CLC number O 211.4 - O 211.6 Foundation item: Supported by the National Natural Science Foundation of China (10071058)Biography: Gan Shi-xin (1939-), male, Professor, research direction: martingale theory, probability limiting theory and Banach space geometry theory.
基金Foundation of Anhui Educational Committee(No.KJ2013Z225)
文摘In the paper,we get the precise results of Hájek-Rényi type inequalities for the partial sums of negatively orthant dependent sequences,which improve the results of Theorem 3.1and Corollary 3.2 in Kim(2006)and the strong law of large numbers and strong growth rate for negatively orthant dependent sequences.
基金National Natural Science Foundation of China! (No. 19701O11) Foundation of "151 talent project" of Zhejiang provience.
文摘By using a Rosenthal type inequality established in this paper, the complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are discussed for *-mixing random fields.
基金Supported by the Natural Science Foundation of Anhui Province(1508085J06) the Key Projects for Academic Talent of Anhui Province(gxbj ZD2016005) the Students Innovative Training Project of Anhui University(201610357001)
文摘In this paper, by the three series theorem of m-negatively associated(m-NA,in short) random variables and the truncation method of random variables, we mainly investigated the strong convergence properties for partial sums of m-NA random variables.In addition, the Khintchine-Kolmogorov convergence theorem and Kolmogorov-type strong law of large numbers for m-NA random variables are also obtained. The results obtained in the paper generalize some corresponding ones for independent random variables and some dependent random variables.
基金Partly supported by the National Natural Science Foundation of China and the Ministry of Education of ChinaPartly supported by the Science and Technology Research Item of Hubei Provincial Department of Education,Jiaghan University
文摘M-negatively associated random variables, which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its particular case, are introduced and studied. Large deviation principles and moderate deviation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.
基金supported by the National Natural Science Foundation of China under Grant Nos.11671012 and 11871072the Natural Science Foundation of Anhui Province under Grant Nos.1808085QA03,1908085QA01,1908085QA07+1 种基金the Provincial Natural Science Research Project of Anhui Colleges under Grant No.KJ2019A0003the Students Innovative Training Project of Anhui University under Grant No.201910357002。
文摘This paper mainly studies the strong convergence properties for weighted sums of extended negatively dependent(END,for short)random variables.Some sufficient conditions to prove the strong law of large numbers for weighted sums of END random variables are provided.In particular,the authors obtain the weighted version of Kolmogorov type strong law of large numbers for END random variables as a product.The results that the authors obtained generalize the corresponding ones for independent random variables and some dependent random variables.As an application,the authors investigate the errors-in-variables(EV,for short)regression models and establish the strong consistency for the least square estimators.Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analysed for illustration.
文摘Let{Xn^-,n^-∈N^d}be a field of Banach space valued random variables, 0 〈r〈p≤2 and{an^-,k^-, (n^-,k^-) ∈ N^d × N^d ,k^-≤n^-} a triangular array of real numhers, where N^d is the d-dimensional lattice (d≥1 ). Under the minimal condition that {||Xn^-|| r,n^- ∈N^d} is {|an^-,k^-|^r,(n^-,k^-)} ∈ N^d ×N^d,k^-≤n^-}-uniformly integrable, we show that ∑(k^-≤n^-)an^-,k^-,Xk^-^(L^r(or a,s,)→0 as |n^-|→∞ In the above, if 0〈r〈1, the random variables are not needed to be independent. If 1≤r〈p≤2, and Banach space valued random variables are independent with mean zero we assume the Banaeh space is of type p. If 1≤r≤p≤2 and Banach space valued random variables are not independent we assume the Banach space is p-smoothable.
基金Supported by the University Students Science Research Training Program of Anhui University(KYXL20110004)
文摘In this paper, we will present some strong convergence results for sequences of ψ-mixing random variables. The results for sequences of ψ-mixing random variables generalize the corresponding results for independent random variable sequences without any extra conditions.
基金Supported by the National Natural Science Foundation of China (Grant No.10571076)
文摘In this paper,we study the strong law of large numbers for the frequencies of occurrence of states and ordered couples of states for nonsymmetric Markov chain(NSMC) indexed by Cayley tree with any finite states.The asymptotic equipartition properties with almost everywhere(a.e.) convergence for NSMC indexed by Cayley tree are obtained.This article generalizes a recent result.
基金Supported by the National Natural Science Foundationof China (10671149)
文摘We mainly study the almost sure limiting behavior of weighted sums of the form ∑ni=1 aiXi/bn , where {Xn, n ≥ 1} is an arbitrary Banach space valued random element sequence or Banach space valued martingale difference sequence and {an, n ≥ 1} and {bn,n ≥ 1} are two sequences of positive constants. Some new strong laws of large numbers for such weighted sums are proved under mild conditions.
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The first author is supported by Basic Science Research Program through the National Research Foundationof Korea funded by the Ministry of Education,Science,and Technology(Grant No.2011-0013791)the secondauthor is partially supported by a grant from the Natural Sciences and Engineering Research Council of Canadathe third author is partially supported by a grant from the Natural Sciences and Engineering Research Councilof Canada
文摘Let (X, Xn; n≥ 1} be a sequence of i.i.d, random variables with values in a measurable space (S,8) such that E|h(X1, X2,..., Xm)| 〈 ∞, where h is a measurable symmetric function from Sm into R = (-∞, ∞). Let {wn,i1,i2 im ; 1 ≤ i1 〈 i2 〈 …… 〈im 〈 n, n ≥ m} be a matrix array of real numbers. Motivated by a result of Choi and Sung (1987), in this note we are concerned with establishing a strong law of large numbers for weighted U-statistics with kernel h of degree m. We show that whenever SUP n≥m max1〈i1〈i2〈…〈im≤|wn i1,i2 i,im| 〈∞, where 0 = Eh(X1, X2,..., Xm). The proof of this result is based on a new general result on complete convergence, which is a fundamental tool, for array of real-valued random variables under some mild conditions.
基金supported by the Vietnam Institute for Advanced Study in Mathematics(VIASM)the Vietnam National Foundation for Sciences and Technology Development NAFOSTED(Grant No.101.01.2012.13)supported by NAFOSTED(Grant No.101.03.2012.17)
文摘For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p(1 ≤ p ≤ 2) Banach spaces.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11071182).
文摘We establish some strong limit theorems for a sequence of pair-wise extended lower/upper negatively dependent random variables and give some new examples of dependent random variables.