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almost sure convergence of the General Jamison Weighted Sum of B-Valued Random Variables

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Acta Mathematica Sinica,English Series 2004年第1期20卷 181-192页

作者： ChunSu DepartmentofStatisticsandFinance UniversityofScienceandTechnologyofchinaHefei230026P.R.China

In this paper,two new functions are introduced to depict the Jamison weighted sum of random variables instead using the common methods,their properties and relationships are system- atically *** also analysed the implication of the conditions in previous *** we apply these consequences to B-valued random variables,and greatly improve the original results of the strong convergence of the general Jamison weighted ***,our discussions are useful to the corresponding questions of real-valued random variables.

关键词： almost sure convergence β-valued random variable General Jamison weighted sum p-smooth Banach space Banach space of type p

almost sure convergence and Complete convergence for the Weighted Sums of Martingale Differences

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Wuhan University Journal of Natural Sciences 1999年第3期4卷 278-284页

作者： Ai-jiao, Deng Jing-jun, Liu College of Mathematical and Computer Sciences Wuhan University Wuhan China

Let {(D n, FFFn),n/->1} be a sequence of martingale differences and {a ni, 1≤i≤n,n≥1} be an array of real constants. almost sure convergence for the row sums ?i = 1n ani D1\sum\limits_{i = 1}^n {a_{ni} D_1 } are discussed. We also discuss complete convergence for the moving average processes underB-valued martingale differences assumption.

关键词： complete convergence almost sure convergence weighted sums, martingale differences moving average processes

almost sure convergence OF THE STABLE TAIL EMPIRICAL DEPENDENCE FUNCTION IN MULTIVARIATE EXTREME STATISTICS

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Acta Mathematicae Applicatae Sinica 1997年第2期13卷 167-175页

作者：祁永成 Department of Probability & Statistics Peking University Beijing China

In this paper we prove the almost sure convergence of the stable tail empirical dependence function for multivariate extreme values.

关键词： Multivariate extreme value stable tail empirical dependence function almost sure convergence

ON almost sure convergence OF WEIGHTED SUMS OF RANDOM ELEMENT SEQUENCES

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Acta Mathematica Scientia 2010年第4期30卷 1021-1028页

作者：甘师信 School of Mathematics and Statistics Wuhan University

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.

关键词： Strong law of large number almost sure convergence Lp convergence weighted sums Banach space valued random element sequence Banach space martingale difference sequence

almost sure convergence of Proximal Stochastic Accelerated Gradient Methods

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Journal of Applied Mathematics and Physics 2024年第4期12卷 1321-1336页

作者： Xin Xiang Haoming Xia Key Laboratory of Optimization Theory and Applications School of Mathematics and Information China West Normal University Nanchong China

Proximal gradient descent and its accelerated version are resultful methods for solving the sum of smooth and non-smooth problems. When the smooth function can be represented as a sum of multiple functions, the stochastic proximal gradient method performs well. However, research on its accelerated version remains unclear. This paper proposes a proximal stochastic accelerated gradient (PSAG) method to address problems involving a combination of smooth and non-smooth components, where the smooth part corresponds to the average of multiple block sums. Simultaneously, most of convergence analyses hold in expectation. To this end, under some mind conditions, we present an almost sure convergence of unbiased gradient estimation in the non-smooth setting. Moreover, we establish that the minimum of the squared gradient mapping norm arbitrarily converges to zero with probability one.

关键词： Proximal Stochastic Accelerated Method almost sure convergence Composite Optimization Non-Smooth Optimization Stochastic Optimization Accelerated Gradient Method

A Supplement to the Baum-Katz-Spitzer Complete convergence Theorem

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Acta Mathematica Sinica,English Series 2007年第3期23卷 557-562页

作者： Andrew ROSALSKY Department of Statistics University of FloridaGainesvilleFlorida 32611USA

Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such thatan↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞Set Sn=∑i=1^n Xi,n≥*** this paper we prove that∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.

关键词： partial sums of i.i.d. Banach space valued random variables Baum-Katz-Spitzer complete convergence theorem almost sure convergence

On the Relationship Between the Baum-Katz-Spitzer Complete convergence Theorem and the Law of the Iterated Logarithm

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Acta Mathematica Sinica,English Series 2007年第4期23卷 599-612页

作者： De Li LI Andrew ROSALSKY Andrei VOLODIN Department o/Mathematical Sciences Lakehead University Thunder Bay ON Canada P7B 5E1 Department of Statistics University of Florida Gainesville FL 32611 USA Department of Mathematics and Statistics University of Regina Regina SK Canada S4S 0A2

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〈∞ *** ∞ ∑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 *** ^∞∑ 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.

关键词： partial sums of i.i.d. Banach space-valued random variables Baum-Katz-Spitzer complete convergence theorem law of the iterated logarithm almost sure convergence

convergence of Jamison-Type Weighted Sums of Pairwise Negatively Quadrant Dependent Random Variables

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Acta Mathematicae Applicatae Sinica 2002年第1期18卷 161-168页

作者： Han-ying LIANG, Zhi-jing Chen, Chun SUDepartment of Applied Mathematics, Tongji University, Shanghai 200092, ChinaDepartment of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China Department of Applied Mathematics Tongji University Shanghai China Department of Statistics and Finance University of Science and Technology of China Hefei China

Under very general weight function, we discuss the convergence of Jamison-type weighted sums of pairwise negatively quadrant dependent (NQD) r.v.'s. The results on i.i.d. setting of [3] and [1] are extended and genera... 详细信息

关键词： Pairwise NQD sequence weighted sum convergence in probability almost sure convergence

THE STRONG LAW OF LARGE NUMBERS FOR PAIRWISE NQD RANDOM VARIABLES

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Journal of Systems Science & Complexity 2011年第2期24卷 347-357页

作者： Qunying WU Yuanying JIANG College of Science Guilin University of Technology Guilin 541004 China.

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.

关键词： almost sure convergence pairwise negatively quadrant dependent random variables,strong law of large numbers.