On Complete Convergence for Weighted Sums of Arrays of Dependent Random Variables

A rate of complete convergence for weighted sums of arrays of rowwise independent random variables was obtained by Sung and Volodin (2011). In this paper, we extend this result to negatively associated and negatively dependent random variables. Similar results for sequences of φ-mixing and ρ*-mixing random variables are also obtained. Our results improve and generalize the results of Baek et al. (2008), Kuczmaszewska (2009), and Wang et al. (2010)

Complete ff-moment convergence for weighted sums of WOD arrays with statistical applications

summary:Complete $f$-moment convergence is much more general than complete convergence and complete moment convergence. In this work, we mainly investigate the complete $f$-moment convergence for weighted sums of widely orthant dependent (WOD, for short) arrays. A general result on Complete $f$-moment convergence is obtained under some suitable conditions, which generalizes the corresponding one in the literature. As an application, we establish the complete consistency for the weighted linear estimator in nonparametric regression models. Finally, some simulations are provided to show the numerical performance of theoretical results based on finite samples

General theorems on exponential and Rosenthal's inequalities and on complete convergence

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On q-Gaussians and Exchangeability

The q-Gaussians are discussed from the point of view of variance mixtures of normals and exchangeability. For each q< 3, there is a q-Gaussian distribution that maximizes the Tsallis entropy under suitable constraints. This paper shows that q-Gaussian random variables can be represented as variance mixtures of normals. These variance mixtures of normals are the attractors in central limit theorems for sequences of exchangeable random variables; thereby, providing a possible model that has been extensively studied in probability theory. The formulation provided has the additional advantage of yielding process versions which are naturally q-Brownian motions. Explicit mixing distributions for q-Gaussians should facilitate applications to areas such as option pricing. The model might provide insight into the study of superstatistics.Comment: 14 page

On Complete Convergence for Weighted Sums of ρ

We prove the strong law of large numbers for weighted sums ∑i=1n‍aniXi, which generalizes and improves the corresponding one for independent and identically distributed random variables and φ-mixing random variables. In addition, we present some results on complete convergence for weighted sums of ρ*-mixing random variables under some suitable conditions, which generalize the corresponding ones for independent random variables

Strong convergence for weighted sums of ρ*-mixing random variables

The authors discuss the strong convergence for weighted sums of sequences of ρ*-mixing random variables. The obtained results extend and improve the corresponding theorem of Bai and Cheng [Bai, Z. D., Cheng, P. E., 2000. Marcinkiewicz strong laws for linear statistics. Statist. Probab. Lett., 46, 105-112]. The method used in this article differs from that of Bai and Cheng (2000)