Unequal virtual water transfer may aggravate local water scarcity risk.However,the quantitative confirmation of a clear geographic convergence between virtual water transfer and water scarcity risk remains undetermine...Unequal virtual water transfer may aggravate local water scarcity risk.However,the quantitative confirmation of a clear geographic convergence between virtual water transfer and water scarcity risk remains undetermined.We present an analytical framework that reveals the spatial matching between global water scarcity risk and virtual water trade inequality.This framework integrates a three-dimensional water scarcity risk assessment,hybrid input-output analysis,pollution trade term construction,and geographic convergence identification.The framework is applied to 123 countries for long-term validation from 1991 to 2021.We show that despite global improvements in water efficiency and security,countries exceeding the maximum water vulnerability threshold have increased by 50%.South Asia is the largest net exporter of virtual water.Central Asia exhibits the most pronounced virtual water trade inequality.To achieve the same economic growth,Central Asia needs to pay several times the local water consumption costs of developed regions(15.9−83.6 times,2021).In the past 30 years,the average geographic convergence index exceeded 0.8.Countries facing severe water scarcity also exhibit pronounced inequalities in virtual water trade,indicating that a significant geographic convergence relationship exists.Effectively responding to this unsustainable relationship necessitates balancing both domestic resource risk management and global virtual water trade regulation.展开更多
Nereididae is a prolific annelid family widely distributed in the world oceans,especially in the Indo-Pacific Convergence Zone(IPCZ).However,its biogeographic pattern remains unexplored in IPCZ.To contribute to the un...Nereididae is a prolific annelid family widely distributed in the world oceans,especially in the Indo-Pacific Convergence Zone(IPCZ).However,its biogeographic pattern remains unexplored in IPCZ.To contribute to the understanding of biodiversity and biogeography of Nereididae in the IPCZ,we integrated historical data of species distributions with those of model-predicted ones to determine the biogeographic patterns of nereid species,from which we projected to its future distribution patterns for 2090-2100 under different climate scenarios(SSP1-1.9 and SSP5-8.5).Functional diversity within IPCZ was assessed using functional richness,functional evenness,and functional disparity.Divergence times within Nereididae were estimated using three DNA marker genes(COI,16S,and 18S rRNA),and a time tree was constructed based on a strict molecular clock model.The IPCZ was established as a key Nereididae biodiversity hotspot through distribution modelling of 256 species(44 genera),and temperature emerging as the predominant climatic driver of species distribution patterns.The distribution of species and functional diversity is notable for its non-centralized pattern.We projected that by the end of the century,areas of medium-to-high species richness will expand significantly under the low-emission SSP1-1.9 climate scenario.However,under the high-emission SSP5-8.5 scenario,the suitability of these regions significantly declines,posing an increasingly severe threat to biodiversity.In addition,by molecular clock analysis,we revealed that the evolutionary divergence of extant nereidid species occurred mainly in the Cretaceous and Jurassic,suggesting that paleogeographical and environmental events,such as oceanic anoxic events,might have played a pivotal role in shaping the evolutionary trajectory and ecological adaptations of marine annelids.These findings highlight the importance of considering both current biodiversity patterns and historical contexts in conservation planning,and provided insights into the potential factors on the biogeographic distribution and evolutionary processes of Nereididae.展开更多
In this paper,we study a comprehensive mathematical model describing the problem of frictional contact between a nonlinear thermo-piezoelectric body and a rigid foundation with electrically conductive effect,in which ...In this paper,we study a comprehensive mathematical model describing the problem of frictional contact between a nonlinear thermo-piezoelectric body and a rigid foundation with electrically conductive effect,in which the contact conditions are described by a Signorini’s condition and Coulomb’s friction law.We derive the variational form of the contact problem which is a mixed system formulated by variational inequalities and equalities.Then,we use standard results on mixed problems and the Banach fixed-point theorem to prove the existence and uniqueness of the solution to the contact problem.Moreover,we demonstrate the convergence of a penalty method for this contact problem under consideration.Finally,finite element method is applied to the penalty contact problem and a strong convergence theorem is obtained.展开更多
In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems includ...In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems including bounded convergence theorem, control convergence theorem and Vitali-type convergence theorem and so on. Moreover, the conditional compactness of its subsets is also discussed.展开更多
We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in ...We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.展开更多
Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Zn（z...Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Zn（z） the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Zn（z） assuming a condition like ＂EN（logN）1＋λ 〈 ∞＂ for the offspring distribution and a finite moment condition on the motion law. This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.展开更多
In this paper, some iterative schemes for approximating the common element of the set of zero points of maximal monotone operators and the set of fixed points of relatively nonexpansive mappings in a real uniformly sm...In this paper, some iterative schemes for approximating the common element of the set of zero points of maximal monotone operators and the set of fixed points of relatively nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Some strong convergence theorems are obtained, to extend the previous work.展开更多
The author shows a characterization of a （unbounded） weakly filter convergent sequence which is parallel to that every weakly null sequence （xn） in a Banach space admits a norm null sequence （yn） with yn ∈ co...The author shows a characterization of a （unbounded） weakly filter convergent sequence which is parallel to that every weakly null sequence （xn） in a Banach space admits a norm null sequence （yn） with yn ∈ co（xk）k≥n for all n ∈ N. A version of the Radon-Riesz type theorem is also proved within the frame of the filter convergence.展开更多
In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-stro...In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces. Then we study the weak convergence of the two iterative sequences. Our results improve and extend the results announced by many others.展开更多
We obtain the expansion of Renyi divergence of order α （0 〈 α 〈 1） between the normalized sum of IID continuous random variables and the Caussian limit under minimal moment conditions via Edgeworth-type expansio...We obtain the expansion of Renyi divergence of order α （0 〈 α 〈 1） between the normalized sum of IID continuous random variables and the Caussian limit under minimal moment conditions via Edgeworth-type expansion. The rate is faster than that of Shannon case, which can be used to improve the rate of convergence in total variance norm.展开更多
Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 ： K → E be two weakly inward nonself asymptotically nonex...Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 ： K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with a sequence {kn^（i）} [1, ∞） （i = 1, 2）, and F ：= F（T1）∩ F（T2） ≠ 0. An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If E has also a Frechet differentiable norm or its dual E^＊ has Kadec-Klee property, then weak convergence theorems are obtained.展开更多
In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some ...In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.展开更多
Let Sigma (infinity)(n=1) X-n be a series of independent random variables with at least one non-degenerate X-n, and let F-n be the distribution function of its partial sums S-n = Sigma (n)(k=1) X-k. Motivated by Hilde...Let Sigma (infinity)(n=1) X-n be a series of independent random variables with at least one non-degenerate X-n, and let F-n be the distribution function of its partial sums S-n = Sigma (n)(k=1) X-k. Motivated by Hildebrand's work in [1], the authors investigate the a.s. convergence of Sigma (infinity)(n=1) X-n under a hypothesis that Sigma (infinity)(n=1) rho (X-n, c(n)) = infinity whener Sigma (infinity)(n=1) c(n) diverges, where the notation rho (X,c) denotes the Levy distance between the random variable X and the constant c. The principal result of this paper shows that the hypothesis is the condition under which the convergence of F-n(x(0)) with the limit value 0 < L-0 < 1, together with the essential convergence of Sigma (infinity)(n=1) X-n, is both sufficient and necessary in order for the series Sigma (infinity)(n=1) X-n to a.s. coverage. Moreover, if the essential convergence of Sigma (infinity)(n=1) X-n is strengthened to limsup(n=infinity) P(\S-n\ < K) = 1 for some K > 0, the hypothesis is already equivalent to the a.s. convergence of Sigma (infinity)(n=1) X-n. Here they have not only founded a very general limit theorem, but improved the related result in Hildebrand([1]) as well.展开更多
In this paper we obtain the uniform bounds on the rate of convergence in the central limit theorem (CLT) for a class of two-parameter martingale difference sequences under certain conditions.
Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to ...Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to f almost everywhere, when t goes to∞. In this paper we use a Theorem established by K.K.Chen to show that if we change their multiplier, then the condition on f can be weakened.展开更多
This paper investigated the convergence of the columns (with fixed denominator degrees) of the multivariate rational interpolations to a meromorphic function. A simple convergence criterion and the rate of convergence...This paper investigated the convergence of the columns (with fixed denominator degrees) of the multivariate rational interpolations to a meromorphic function. A simple convergence criterion and the rate of convergence were given. The criterion is a nature extension of the theorem of Saff for the convergence of columns of univariate rational interpolations.展开更多
Practical sufficient conditions for the convergence of the AOR method and a practical sufficient condition for H-matrices are studied. The obtained convergence conditions suited to matrices, which need not to be diago...Practical sufficient conditions for the convergence of the AOR method and a practical sufficient condition for H-matrices are studied. The obtained convergence conditions suited to matrices, which need not to be diagonally dominant.展开更多
In this article, we study two types of martingale ergodic processes. We prove that a.e. convergence and L^p convergence as well as maximal inequalities, which are established both in ergodic theory and martingale sett...In this article, we study two types of martingale ergodic processes. We prove that a.e. convergence and L^p convergence as well as maximal inequalities, which are established both in ergodic theory and martingale setting, also hold well for these new sequences of random variables. Moreover, the corresponding theorems in the former two areas turn out to be degenerate cases of the martingale ergodic theorems proved here.展开更多
Under the conditions on covariances of the original random variables, a Glivenko-Cantelli theorem for associated sequences and weak convergence for empirical processes of stationary associated sequences are obtained, ...Under the conditions on covariances of the original random variables, a Glivenko-Cantelli theorem for associated sequences and weak convergence for empirical processes of stationary associated sequences are obtained, assuming the random variables to be discrete.展开更多
基金supported by National Natural Science Foundation of China(Grant No.52279027)National Key R&D Program of China(Grant No.2021YFC3200201)+1 种基金Key Research Project on Decision Consultation of the Strategic Development Department of China Association for Science and Technology(Grant No.2023070615CG111504)China Engineering Science and Technology Development Strategy Henan Research Institute Strategic Consulting Research Project(Grant No.2024HENYB01).
文摘Unequal virtual water transfer may aggravate local water scarcity risk.However,the quantitative confirmation of a clear geographic convergence between virtual water transfer and water scarcity risk remains undetermined.We present an analytical framework that reveals the spatial matching between global water scarcity risk and virtual water trade inequality.This framework integrates a three-dimensional water scarcity risk assessment,hybrid input-output analysis,pollution trade term construction,and geographic convergence identification.The framework is applied to 123 countries for long-term validation from 1991 to 2021.We show that despite global improvements in water efficiency and security,countries exceeding the maximum water vulnerability threshold have increased by 50%.South Asia is the largest net exporter of virtual water.Central Asia exhibits the most pronounced virtual water trade inequality.To achieve the same economic growth,Central Asia needs to pay several times the local water consumption costs of developed regions(15.9−83.6 times,2021).In the past 30 years,the average geographic convergence index exceeded 0.8.Countries facing severe water scarcity also exhibit pronounced inequalities in virtual water trade,indicating that a significant geographic convergence relationship exists.Effectively responding to this unsustainable relationship necessitates balancing both domestic resource risk management and global virtual water trade regulation.
基金Supported by the Strategic Priority Research Program of the Chinese Academy of Sciences(No.XDB42000000)the National Natural Science Foundation of China(No.42376092)the Marine S&T Fund of Shandong Province for Pilot National Laboratory for Marine Science and Technology(No.2022QNLM030004)。
文摘Nereididae is a prolific annelid family widely distributed in the world oceans,especially in the Indo-Pacific Convergence Zone(IPCZ).However,its biogeographic pattern remains unexplored in IPCZ.To contribute to the understanding of biodiversity and biogeography of Nereididae in the IPCZ,we integrated historical data of species distributions with those of model-predicted ones to determine the biogeographic patterns of nereid species,from which we projected to its future distribution patterns for 2090-2100 under different climate scenarios(SSP1-1.9 and SSP5-8.5).Functional diversity within IPCZ was assessed using functional richness,functional evenness,and functional disparity.Divergence times within Nereididae were estimated using three DNA marker genes(COI,16S,and 18S rRNA),and a time tree was constructed based on a strict molecular clock model.The IPCZ was established as a key Nereididae biodiversity hotspot through distribution modelling of 256 species(44 genera),and temperature emerging as the predominant climatic driver of species distribution patterns.The distribution of species and functional diversity is notable for its non-centralized pattern.We projected that by the end of the century,areas of medium-to-high species richness will expand significantly under the low-emission SSP1-1.9 climate scenario.However,under the high-emission SSP5-8.5 scenario,the suitability of these regions significantly declines,posing an increasingly severe threat to biodiversity.In addition,by molecular clock analysis,we revealed that the evolutionary divergence of extant nereidid species occurred mainly in the Cretaceous and Jurassic,suggesting that paleogeographical and environmental events,such as oceanic anoxic events,might have played a pivotal role in shaping the evolutionary trajectory and ecological adaptations of marine annelids.These findings highlight the importance of considering both current biodiversity patterns and historical contexts in conservation planning,and provided insights into the potential factors on the biogeographic distribution and evolutionary processes of Nereididae.
基金supported by the Project for Outstanding Young Talents in Bagui of Guangxi,the Natural Science Foundation of Guangxi(2021GXNSFFA196004,2024GXNSFBA010337)the NSFC(12371312)+2 种基金the Natural Science Foundation of Chongqing(CSTB2024NSCQ-JQX0033)supported by the Postdoctoral Fellowship Program of CPSF(GZC20241534)the Startup Project of Postdoctoral Scientific Research of Zhejiang Normal University(ZC304023924).
文摘In this paper,we study a comprehensive mathematical model describing the problem of frictional contact between a nonlinear thermo-piezoelectric body and a rigid foundation with electrically conductive effect,in which the contact conditions are described by a Signorini’s condition and Coulomb’s friction law.We derive the variational form of the contact problem which is a mixed system formulated by variational inequalities and equalities.Then,we use standard results on mixed problems and the Banach fixed-point theorem to prove the existence and uniqueness of the solution to the contact problem.Moreover,we demonstrate the convergence of a penalty method for this contact problem under consideration.Finally,finite element method is applied to the penalty contact problem and a strong convergence theorem is obtained.
基金Supported by Hubei Research Center for Financial Development and Financial Security(2008D029)
文摘In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems including bounded convergence theorem, control convergence theorem and Vitali-type convergence theorem and so on. Moreover, the conditional compactness of its subsets is also discussed.
基金supported by the National Natural Science Foundation of China(11571052,11731012)the Hunan Provincial Natural Science Foundation of China(2018JJ2417)the Open Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering(2018MMAEZD02)。
文摘We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.
文摘Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Zn（z） the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Zn（z） assuming a condition like ＂EN（logN）1＋λ 〈 ∞＂ for the offspring distribution and a finite moment condition on the motion law. This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.
基金the National Natural Science Foundation of China (10771050)
文摘In this paper, some iterative schemes for approximating the common element of the set of zero points of maximal monotone operators and the set of fixed points of relatively nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Some strong convergence theorems are obtained, to extend the previous work.
基金partially supported by the Natural Science Foundation of China(11426061,11501108)the Natural Science Foundation of Fujian province(2015J01579)
文摘The author shows a characterization of a （unbounded） weakly filter convergent sequence which is parallel to that every weakly null sequence （xn） in a Banach space admits a norm null sequence （yn） with yn ∈ co（xk）k≥n for all n ∈ N. A version of the Radon-Riesz type theorem is also proved within the frame of the filter convergence.
文摘In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces. Then we study the weak convergence of the two iterative sequences. Our results improve and extend the results announced by many others.
基金supported by National Basic Research Program of China(973 Program)(2011CB707802,2013CB910200)Natural Science Foundation of China Grant(11126180)
文摘We obtain the expansion of Renyi divergence of order α （0 〈 α 〈 1） between the normalized sum of IID continuous random variables and the Caussian limit under minimal moment conditions via Edgeworth-type expansion. The rate is faster than that of Shannon case, which can be used to improve the rate of convergence in total variance norm.
文摘Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 ： K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with a sequence {kn^（i）} [1, ∞） （i = 1, 2）, and F ：= F（T1）∩ F（T2） ≠ 0. An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If E has also a Frechet differentiable norm or its dual E^＊ has Kadec-Klee property, then weak convergence theorems are obtained.
基金Supported in part by the National Social Science Foundation of China(19BTJ020)。
文摘In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.
文摘Let Sigma (infinity)(n=1) X-n be a series of independent random variables with at least one non-degenerate X-n, and let F-n be the distribution function of its partial sums S-n = Sigma (n)(k=1) X-k. Motivated by Hildebrand's work in [1], the authors investigate the a.s. convergence of Sigma (infinity)(n=1) X-n under a hypothesis that Sigma (infinity)(n=1) rho (X-n, c(n)) = infinity whener Sigma (infinity)(n=1) c(n) diverges, where the notation rho (X,c) denotes the Levy distance between the random variable X and the constant c. The principal result of this paper shows that the hypothesis is the condition under which the convergence of F-n(x(0)) with the limit value 0 < L-0 < 1, together with the essential convergence of Sigma (infinity)(n=1) X-n, is both sufficient and necessary in order for the series Sigma (infinity)(n=1) X-n to a.s. coverage. Moreover, if the essential convergence of Sigma (infinity)(n=1) X-n is strengthened to limsup(n=infinity) P(\S-n\ < K) = 1 for some K > 0, the hypothesis is already equivalent to the a.s. convergence of Sigma (infinity)(n=1) X-n. Here they have not only founded a very general limit theorem, but improved the related result in Hildebrand([1]) as well.
文摘In this paper we obtain the uniform bounds on the rate of convergence in the central limit theorem (CLT) for a class of two-parameter martingale difference sequences under certain conditions.
基金Supported by the National Natural Science Foundation of China(11071065,11171306)
文摘Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to f almost everywhere, when t goes to∞. In this paper we use a Theorem established by K.K.Chen to show that if we change their multiplier, then the condition on f can be weakened.
文摘This paper investigated the convergence of the columns (with fixed denominator degrees) of the multivariate rational interpolations to a meromorphic function. A simple convergence criterion and the rate of convergence were given. The criterion is a nature extension of the theorem of Saff for the convergence of columns of univariate rational interpolations.
文摘Practical sufficient conditions for the convergence of the AOR method and a practical sufficient condition for H-matrices are studied. The obtained convergence conditions suited to matrices, which need not to be diagonally dominant.
文摘In this article, we study two types of martingale ergodic processes. We prove that a.e. convergence and L^p convergence as well as maximal inequalities, which are established both in ergodic theory and martingale setting, also hold well for these new sequences of random variables. Moreover, the corresponding theorems in the former two areas turn out to be degenerate cases of the martingale ergodic theorems proved here.
文摘Under the conditions on covariances of the original random variables, a Glivenko-Cantelli theorem for associated sequences and weak convergence for empirical processes of stationary associated sequences are obtained, assuming the random variables to be discrete.
