Let $ \{X_n, n\geq 1\} $ be a sequence of independent and identically distributed random variables in a sublinear expectation $ (\Omega, \mathcal H, {\mathbb {\widehat{E}}}) $ with a capacity $ {\mathbb V} $ under $ {\mathbb {\widehat{E}}} $. In this paper, under some suitable conditions, I show that a general form of precise asymptotics for complete convergence holds under sublinear expectation. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in studies of complete convergence. The results extend some precise asymptotics for complete convergence theorems from the traditional probability space to the sublinear expectation space. The results also generalize the known results obtained by Xu and Cheng ^[34].

In this article, the complete moment convergence for the partial sum of moving average processes { X n = ∑ i = − ∞ ∞ a i Y i + n , n ≥ 1 } $\{X_{n}=\sum_{i=-\infty}^{\infty}a_{i}Y_{i+n},n\geq 1\}$ is established under some mild conditions, where { Y i , − ∞ < i < ∞ } $\{Y_{i},-\infty < i<\infty\}$ is a doubly infinite sequence of random variables satisfying the Rosenthal type maximal inequality and { a i , − ∞ < i < ∞ } $\{a_{i},-\infty< i<\infty\}$ is an absolutely summable sequence of real numbers. These conclusions promote and improve the corresponding results given by Ko (J. Inequal. Appl. 2015:225, 2015).

In this paper, we study the limit properties of the ratio for order statistics based on samples from an exponential distribution and obtain the expression of the density functions, the existence of the moments, the strong law of large numbers for R n i j $R_{nij}$ with 1 ≤ i < j < m n = m $1\leq i< j< m_{n}=m$ . We also discuss other limit theorems such as the central limit theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure central limit theorem for self-normalized sums of R n i j $R_{nij}$ with 2 ≤ i < j < m n = m $2\leq i< j< m_{n}=m$ .