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The Marcinkiewicz–Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences
Tác giả: Vũ T Ngọc Ánh, Nguyễn T Thanh Hiền, Lê V Thành, Võ T Hồng Vân
429    1
Journal of Theoretical Probability
Quyển: 34/no.1     Trang: 331--348
Đường link/DOI: https://link.springer.com/article/10.1007%2Fs10959-019-00973-2
Minh chứng: 1523-2021-jotp-official.pdf
Năm xuất bản: 1/2021
Tóm tắt
This paper establishes complete convergenceforweighted sums and theMarcinkiewicz--Zygmund-type strong law of large numbers for sequences of negatively associated and identically distributed random variables {X, Xn, n ≥ 1} with general normalizing constants under a moment condition that E R(X) < ∞, where R(·) is a regularly varying function. The result is new even when the random variables are independent and identically distributed (i.i.d.), and a special case of this result comes close to a solution to an open question raised by Chen and Sung (Stat Probab Lett 92:45–52, 2014). The proof exploits some properties of slowly varying functions and the de Bruijn conjugates. A counterpart of the main result obtained by Martikainen (J Math Sci 75(5):1944--1946, 1995) on the Marcinkiewicz--Zygmund-type strong law of large numbers for pairwise i.i.d. random variables is also presented. Two illustrative examples are provided, including a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game.
Từ khóa
Weighted sum · Negative association · Negative dependence · Complete convergence · Strong law of large numbers · Normalizing constant · Slowly varying function
Cùng tác giả
On the Laws of Large Numbers for Double Arrays of Independent Random Elements in Banach SpacesWeighted sums of strongly mixing random variables with an application to nonparametric regressionOn the weak laws of large numbers for sums of negatively associated random vectors in Hilbert spacesOn complete convergence in mean for double sums of independent random elements in Banach spacesHien, Nguyen Thi Thanh; Thanh, Le Van; Van, Vo Thi Hong. On the negative dependence in Hilbert spaces with applications. Appl. Math. 64 (2019), no. 1, 45--59.Le Van Thanh and Nguyen Thi Thuy, Necessary and sufficient conditions for complete convergence of double weighted sums of pairwise independent identically distributed random elements in Banach spaces, Acta Mathematica Hungarica, Volume 157 (2019), Issue 2, 312–326 (SCIE).Lê Vǎn Thành and Nguyen Ngoc Tu, Error bounds in normal approximation for the squared-length of total spin in the mean field classical N-vector models, Electronic Communications in Probability, Volume 24 (2019) no. 16, 12 pp (SCIE).Bất đẳng thức Berry-Esseen dạng không đềuOn the Baum-Katz theorem for sequences of pairwise independent random variables with regularly varying normalizing constantsVề một sự mở rộng của bổ đề Borel-Cantelli đối với mảng hai chiều các biến cố phụ thuộcOn convergence in mean for double arrays of pairwise independent random variablesOn the error bound in the normal approximation for Jack measuresA note on the stochastic domination condition and uniform integrability with applications to the strong law of large numbersThe Hsu-Robbins-Erdös theorem for the maximum partial sums of quadruplewise independent random variablesOn the (p,q)-type strong law of large numbers for sequences of independent random variablesOn a new concept of stochastic domination and the laws of large numbersOn weak laws of large numbers for maximal partial sums of pairwise independent random variablesOptimal bounds in normal approximation for many interacting worldsOptimal moment conditions for complete convergence for maximal normed weighted sums from arrays of rowwise independent random elements in Banach spacesMean convergence theorems for arrays of dependent random variables with applications to dependent bootstrap and non-homogeneous Markov chainsAlmost sure summability of the maximal normed partial sums of m-dependent random elements in Banach spacesSharp sufficient conditions for mean convergence of the maximal partial sums of dependent random variables with general norming sequencesOn Rio's proof of limit theorems for dependent random fieldsComplete convergence for the maximal partial sums without maximal inequalitiesMaximal inequalities for normed double sums of random elements in martingale type p Banach spaces with applications to degenerate mean convergence of the maximum of normed sumsSome Mean Convergence Theorems for the Maximum of Normed Double Sums of Banach Space Valued Random ElementsLaws of large numbers for pairwise independent random variablesGiáo trình xác suất và thống kê