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

Let {Xnk, k ⩾ 1, n ⩾ 1} be an array of rowwise ANA random variables stochastically dominated by a random variable X, and {ank, 1 ⩽ k ⩽ n, n ⩾ 1} be an array of real numbers with ∑k=1n|ank|α=O(n)$\sum \nolimits _{k=1}^{n}|a_{nk}|^{\alpha }=O(n)$ for some 0 ⩽ α ⩽ 2. Under the almost optimal moment conditions, the paper shows that ∑n=1∞n-1Pmax1≤m≤n∑k=1mankXnk>εn1α(logn)1γ 0. \begin{equation*} \sum \limits _{n=1}^{\infty }n^{-1}P\left\lbrace \max \limits _{1\le m\le n}\left|\sum \limits _{k=1}^{m}a_{nk}X_{nk}\right|>\varepsilon n^{\frac{1}{\alpha }}(\log n)^{\frac{1}{\gamma }}\right\rbrace 0. \end{equation*} In addition, we can get the strong law of large numbers for sequence of ANA random variables as application of our Theorem.