Sufficient and necessary conditions of convergence for ρ͠ mixing random variables

Shuili Zhang ,Cong Qu,Yu Miao

doi:10.1515/math-2019-0036

Shuili Zhang , Cong Qu + Show 1 more

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https://doi.org/10.1515/math-2019-0036

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Abstract

Abstract In the present paper, the sufficient and necessary conditions of the complete convergence and complete moment convergence for ρ͠-mixing random variables are established, which extend some well-known results.

Highlights

In the present paper, the su cient and necessary conditions of the complete convergence and complete moment convergence for ρ-mixing random variables are established, which extend some wellknown results
The notion of ρ-mixing random variables was rst introduced by Bradley [1], and a number of limits results for ρ-mixing random variables have been established by many authors
One can refer to Bradley [1] for the central limit theorem; Sung [2, 3], An and Yuan [4], Lan [5], Guo and Zhu [6] for complete convergence; Zhang [7] for complete moment convergence; Peligrad and Gut [8], Utev and Peligrad [9] for the moment inequalities; Gan [10], Wu and Jiang [11], Kuczmaszewska [12] for strong law of large numbers

Summary

Introduction

Let (Ω, F, P) be a probability space, {Xn , n ≥ } be a sequence of random variables de ned on (Ω, F, P), n. De ne the ρ-mixing coe cients by ρn = sup{ρ(FS , FT) : S, T ⊂ N with dist(S, T) ≥ n},. The sequence {Xn , n ≥ } is called ρ-mixing if there exists k ∈ N such that ρk

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Main results

Proofs of Main results

Xi ε n α