On convergence rate for weighted sums of arrays of rowwise ANA random variables

ABSTRACT

Let {X_nk, k ⩾ 1, n ⩾ 1} be an array of rowwise ANA random variables stochastically dominated by a random variable X, and {a_nk, 1 ⩽ k ⩽ n, n ⩾ 1} be an array of real numbers with ${\sum }_{k=1}^n{\left|a_{nk}\right|}^{\alpha }=O\left(n\right)$ for some 0 ⩽ α ⩽ 2. Under the almost optimal moment conditions, the paper shows that $$\sum _{n=1}^{\infty }{n}^{-1}P\left\{\underset{1\le m\le n}{max}\left|\sum _{k=1}^m{a}_{nk}{X}_{nk}\right|>\epsilon n^{\frac{1}{\alpha }}{(logn)}^{\frac{1}{\gamma }}\right\}<\infty ,\forall \epsilon >0.$$In addition, we can get the strong law of large numbers for sequence of ANA random variables as application of our Theorem.

KEYWORDS:

• Array of rowwise ANA random variables
• complete convergence
• weighted sums

MATHEMATICS SUBJECT CLASSIFICATION:

• 60E15
• 60F15

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (11671012, 11501005, 11701004), Natural Science Foundation for Colleges and Universitiesof Anhui Province (KJ2015A065, KJ2016A027, KJ2017A027), Natural Science Foundation of Anhui Province (1508085J06, 1608085QA02), Students Science Research Training Program of Anhui University (KYXL2016007, KYXL2017005).