In this paper,we study the strong law of large numbers and Shannon-McMillan (S-M) theorem for Markov chains indexed by an infinite tree with uniformly bounded degree.The results generalize the analogous results on a h...In this paper,we study the strong law of large numbers and Shannon-McMillan (S-M) theorem for Markov chains indexed by an infinite tree with uniformly bounded degree.The results generalize the analogous results on a homogeneous tree.展开更多
We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed(IID) random variables for sub-linear expectations initiated by Peng.It turns out...We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed(IID) random variables for sub-linear expectations initiated by Peng.It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.展开更多
Both residual Cesaro alpha-integrability （RCI（α） and strongly residual Cesaro alpha- integrability （SRCI（α）） are two special kinds of extensions to uniform integrability, and both asymptotically almost negati...Both residual Cesaro alpha-integrability （RCI（α） and strongly residual Cesaro alpha- integrability （SRCI（α）） are two special kinds of extensions to uniform integrability, and both asymptotically almost negative association （AANA） and asymptotically quadrant sub-independence （AQSI） are two special kinds of dependence structures. By relating the RCI（α） property as well as the SRCI（α） property with dependence condition AANA or AQSI, we formulate some tail-integrability conditions under which for appropriate α the RCI（（α） property yields Ll-convergence results and the SRCI（α） property yields strong laws of large numbers, which is the continuation of the corresponding literature.展开更多
Let{X_(ni),F_(ni);1≤i≤n,n≥1}be an array of R^(d)martingale difference random vectors and{A_(ni),1≤i≤n,n≥1}be an array of m×d matrices of real numbers.In this paper,the Marcinkiewicz-Zygmund type weak law of...Let{X_(ni),F_(ni);1≤i≤n,n≥1}be an array of R^(d)martingale difference random vectors and{A_(ni),1≤i≤n,n≥1}be an array of m×d matrices of real numbers.In this paper,the Marcinkiewicz-Zygmund type weak law of large numbers for maximal weighted sums of martingale difference random vectors is obtained with not necessarily finite p-th(1<p<2)moments.Moreover,the complete convergence and strong law of large numbers are established under some mild conditions.An application to multivariate simple linear regression model is also provided.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, strong laws of large numbers for weighted sums of ■-mixing sequence are investigated. Our results extend the corresponding results for negatively associated sequence to the case of ■-mixing sequence.
In this paper, the almost sure convergence for pairwise negatively quadrant dependent random variables is studied. The strong law of large numbers for pairwise negatively quadrant dependent random variables is obtaine...In this paper, the almost sure convergence for pairwise negatively quadrant dependent random variables is studied. The strong law of large numbers for pairwise negatively quadrant dependent random variables is obtained. Our results generalize and improve those on almost sure convergence theorems previously obtained by Marcinkiewicz （1937）, Jamison （1965）, Matula （1992） and Wu （2001） from the independent identically distributed （i.i.d.） case to pairwise NQD sequences.展开更多
Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As app...Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As application, some strong laws of large numbers are given. For the case of geometrically decreasing covariances, we obtain the rate of convergence n-1/2(log log n)1/2(logn) which is close to the optimal achievable convergence rate for independent random variables under an iterated logarithm, while Ioannides and Roussas (1999), and Oliveira (2005) only got n-1/3(logn)2/3 and n-1/3(logn)5/3, separately.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
This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables,the author obtains a strong law of large numbers under...This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables,the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite. By discussing the relation between sublinear expectation and Choquet expectation, for a sequence of i.i.d random variables, the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.展开更多
In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence ...In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on R^(∞) under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given.展开更多
This note is devoted to introduce a new concept of conditionally dominated random variables.Under suitable restrict conditions,a general strong law of large numbers for arbitrary continuous random variables is obtained.
Given n samples(viewed as an n-tuple)of aγ-regular discrete distributionπ,in this article the authors concern with the weighted and unweighted graphs induced by the n samples.They first prove a series of SLLN result...Given n samples(viewed as an n-tuple)of aγ-regular discrete distributionπ,in this article the authors concern with the weighted and unweighted graphs induced by the n samples.They first prove a series of SLLN results(of Dvoretzky-Erdos'type).Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1+γThey also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distributions with constant exponent 2.This exponent is obviously independent of the parameterγ∈(0,1),which is a surprise to us at first sight.展开更多
In this paper, we give some conditions on diverging rate of series of the probabilities and converging rate of series of the α-mixing coefficients for sequences of events, under which the conclusion of the Second Bor...In this paper, we give some conditions on diverging rate of series of the probabilities and converging rate of series of the α-mixing coefficients for sequences of events, under which the conclusion of the Second Borel-Cantelli Lemma holds. As corollaries, some moment conditions are obtained, under which the strong law of large numbers holds for sequences of identically distributed random variables.展开更多
Based on fuzzy random variables, the concept of fuzzy stochastic sequences is defined. Strong limit theorems for fuzzy stochastic sequences are established. Some known results in non-fuzzy stochastic sequences are ext...Based on fuzzy random variables, the concept of fuzzy stochastic sequences is defined. Strong limit theorems for fuzzy stochastic sequences are established. Some known results in non-fuzzy stochastic sequences are extended. In order to prove results of this paper, the notion of fuzzy martingale difference sequences is also introduced.展开更多
We investigate a tagged particle in the exclusion processes on {1,..., N }×Zd, with different densities in different levels {k} × Zd, ? k. Ignoring the level the tagged particle lying in, we only concern its...We investigate a tagged particle in the exclusion processes on {1,..., N }×Zd, with different densities in different levels {k} × Zd, ? k. Ignoring the level the tagged particle lying in, we only concern its position in Zd,denoted by Xt. Note that the whole space is not homogeneous. We define the environment process viewed from the tagged particle, of which Xt can be expressed as a functional. It is called the tagged particle process. We show the ergodicity of the tagged particle process, then prove the strong law of large numbers. Furthermore, we show the central limit theorem of Xt provided the zero-mean condition.展开更多
For left truncated and right censored model, letF n be the product-limit estimate and φ a nonnegative measurable function. The almost sure limits of the cumulative hazard function based onF n pd the integral ∫ ?dF n...For left truncated and right censored model, letF n be the product-limit estimate and φ a nonnegative measurable function. The almost sure limits of the cumulative hazard function based onF n pd the integral ∫ ?dF n are given. The results are useful in establishing strong consistent results of various estimates. For left truncated data, similar results were obtained in literature.展开更多
基金the National Natural Science Foundation of China (Grant No.10571076)
文摘In this paper,we study the strong law of large numbers and Shannon-McMillan (S-M) theorem for Markov chains indexed by an infinite tree with uniformly bounded degree.The results generalize the analogous results on a homogeneous tree.
基金supported by National Natural Science Foundation of China(Grant No.11231005)
文摘We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed(IID) random variables for sub-linear expectations initiated by Peng.It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.
基金Supported by National Natural Science Foundation of China (Grant No. 10871217)Natural Science Foundation Project of CQ CSTC of China (Grant No. 2009BB2370)SCR of Chongqing Municipal Education Commission (Grant Nos. KJ090703, KJ100726)
文摘Both residual Cesaro alpha-integrability （RCI（α） and strongly residual Cesaro alpha- integrability （SRCI（α）） are two special kinds of extensions to uniform integrability, and both asymptotically almost negative association （AANA） and asymptotically quadrant sub-independence （AQSI） are two special kinds of dependence structures. By relating the RCI（α） property as well as the SRCI（α） property with dependence condition AANA or AQSI, we formulate some tail-integrability conditions under which for appropriate α the RCI（（α） property yields Ll-convergence results and the SRCI（α） property yields strong laws of large numbers, which is the continuation of the corresponding literature.
基金Supported by the Outstanding Youth Research Project of Anhui Colleges(Grant No.2022AH030156)。
文摘Let{X_(ni),F_(ni);1≤i≤n,n≥1}be an array of R^(d)martingale difference random vectors and{A_(ni),1≤i≤n,n≥1}be an array of m×d matrices of real numbers.In this paper,the Marcinkiewicz-Zygmund type weak law of large numbers for maximal weighted sums of martingale difference random vectors is obtained with not necessarily finite p-th(1<p<2)moments.Moreover,the complete convergence and strong law of large numbers are established under some mild conditions.An application to multivariate simple linear regression model is also provided.
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Foundation item： Supported by the National Natural Science Foundation of China（11171001, 11201001） Supported by the Natural Science Foundation of Anhui Province（t208085QA03, 1308085QA03）
文摘In this paper, strong laws of large numbers for weighted sums of ■-mixing sequence are investigated. Our results extend the corresponding results for negatively associated sequence to the case of ■-mixing sequence.
基金This research is supported by the National Natural Science Foundation of China under Grant No. 11061012, the Support Program of the New Century Guangxi China Ten-hundred-thousand Talents Project under Grant No. 2005214, and the Guangxi, China Science Foundation under Grant No. 2010GXNSFA013120.
文摘In this paper, the almost sure convergence for pairwise negatively quadrant dependent random variables is studied. The strong law of large numbers for pairwise negatively quadrant dependent random variables is obtained. Our results generalize and improve those on almost sure convergence theorems previously obtained by Marcinkiewicz （1937）, Jamison （1965）, Matula （1992） and Wu （2001） from the independent identically distributed （i.i.d.） case to pairwise NQD sequences.
基金the National Natural Science Fbundation of China (Grant Nos. 10161004, 70221001, 70331001)the Natural Science Foundation of Guangxi Province of China (Grant No. 04047033)
文摘Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As application, some strong laws of large numbers are given. For the case of geometrically decreasing covariances, we obtain the rate of convergence n-1/2(log log n)1/2(logn) which is close to the optimal achievable convergence rate for independent random variables under an iterated logarithm, while Ioannides and Roussas (1999), and Oliveira (2005) only got n-1/3(logn)2/3 and n-1/3(logn)5/3, separately.
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.
基金supported by the National Natural Science Foundation of China(Nos.11501325,11231005)
文摘This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables,the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite. By discussing the relation between sublinear expectation and Choquet expectation, for a sequence of i.i.d random variables, the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.
基金Supported by grants from the NSF of China(Grant Nos.11731012,12031005)Ten Thousands Talents Plan of Zhejiang Province(Grant No.2018R52042)+1 种基金NSF of Zhejiang Province(Grant No.LZ21A010002)the Fundamental Research Funds for the Central Universities。
文摘In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on R^(∞) under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given.
基金Supported by the National Nature Science Foundation of China（10571076） Supported by Anhui High Education Research（2006Kj246B）
文摘This note is devoted to introduce a new concept of conditionally dominated random variables.Under suitable restrict conditions,a general strong law of large numbers for arbitrary continuous random variables is obtained.
基金This work was supported by the National Natural Science Foundation of China(Nos.11871162,11771286,11271255,11271077,11001173,11790273)the Key Laboratory of Mathematics for Non linear Science,Fudan University.
文摘Given n samples(viewed as an n-tuple)of aγ-regular discrete distributionπ,in this article the authors concern with the weighted and unweighted graphs induced by the n samples.They first prove a series of SLLN results(of Dvoretzky-Erdos'type).Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1+γThey also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distributions with constant exponent 2.This exponent is obviously independent of the parameterγ∈(0,1),which is a surprise to us at first sight.
基金Supported by the SCR of Chongqing Municipal Education Commission(KJ090703)
文摘In this paper, we give some conditions on diverging rate of series of the probabilities and converging rate of series of the α-mixing coefficients for sequences of events, under which the conclusion of the Second Borel-Cantelli Lemma holds. As corollaries, some moment conditions are obtained, under which the strong law of large numbers holds for sequences of identically distributed random variables.
基金Supported by National Basic Research Programof China (973Program, No.2007CB814901)Research Funds for Doctorial Programs of Higher Education (No.20060255006)Anhui Natural Science Foundation of University (No. KJ2008B143)
文摘Based on fuzzy random variables, the concept of fuzzy stochastic sequences is defined. Strong limit theorems for fuzzy stochastic sequences are established. Some known results in non-fuzzy stochastic sequences are extended. In order to prove results of this paper, the notion of fuzzy martingale difference sequences is also introduced.
基金supported by National Natural Science Foundation of China(Grant No.11371040)
文摘We investigate a tagged particle in the exclusion processes on {1,..., N }×Zd, with different densities in different levels {k} × Zd, ? k. Ignoring the level the tagged particle lying in, we only concern its position in Zd,denoted by Xt. Note that the whole space is not homogeneous. We define the environment process viewed from the tagged particle, of which Xt can be expressed as a functional. It is called the tagged particle process. We show the ergodicity of the tagged particle process, then prove the strong law of large numbers. Furthermore, we show the central limit theorem of Xt provided the zero-mean condition.
基金the National Natural Science Foundation of China (Grant No. 19971006) .
文摘For left truncated and right censored model, letF n be the product-limit estimate and φ a nonnegative measurable function. The almost sure limits of the cumulative hazard function based onF n pd the integral ∫ ?dF n are given. The results are useful in establishing strong consistent results of various estimates. For left truncated data, similar results were obtained in literature.
