Sublinear Expectation Space Research Articles

Marcinkiewicz strong laws for weighted sums under the sub-linear expectations

In this article, we establish complete convergence theorems for weighted sums in sub-linear expectations space. As corollaries, we obtain Marcinkiewicz strong laws for weighted sums under the sublinear expectations. Our results extend and improve the corresponding results of Bai and Cheng (Statist. Probab. Lett. 46: 105-112, 2000) and Cuzick (J. Theoret. Probab. 8: 625-641, 1995) from classical probability space to sub-linear expectation space.

Complete integral convergence for weighted sums of negatively dependent random variables under sub-linear expectations

<abstract><p>In the paper, the complete convergence and complete integral convergence for weighted sums of negatively dependent random variables under the sub-linear expectations are established. The results in the paper extend some complete moment convergence theorems from the classical probability space to the situation of sub-linear expectation space.</p></abstract>

Complete convergence and complete integral convergence for weighted sums of widely negative dependent random variables under the sub-linear expectations

In this work, we work on the complete convergence and complete integral convergence for weighted sums for arrays of row-wise widely negative dependent random variables under sub-linear expectation space, promoting the corresponding results from conventional probability space to sub-linear expectation space. This result extends the corresponding outcome obtained by Liang and Wu (2019).

The laws of large numbers for Pareto-type random variables under sub-linear expectation

In this paper, some laws of large numbers are established for random variables that satisfy the Pareto distribution, so that the relevant conclusions in the traditional probability space are extended to the sub-linear expectation space. Based on the Pareto distribution, we obtain the weak law of large numbers and strong law of large numbers of the weighted sum of some independent random variable sequences.

The moments of the maximum of normalized partial sums related to laws of the iterated logarithm under the sub-linear expectation

Let {Xn;n≥1} be a sequence of independent and identically distributed random variables on a sub-linear expectation space (Ω,ℋ,Ê), Sn=X1+…+Xn. We consider the moments of maxn≥1|Sn|/2nloglogn. The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.

Complete integral convergence for weighted sums of widely negative dependent random variables under the sub-linear expectations

In this article, we establish some results on the complete convergence and complete integral convergence for weighted sums of widely negative dependent random variables under the sub-linear expectations, which extend some complete moment convergence theorems from the classical probability space to the situation of sub-linear expectation space.

Complete convergence theorems for arrays of row-wise extended negatively dependent random variables under sub-linear expectations

The goal of this paper is to establish complete convergence theorems for an array of row-wise END random variables under sub-linear expectation space. As applications of the exponential inequalities, we extend some complete convergence theorems from the traditional probability space to the sub-linear expectation space and our results generalize corresponding results obtained by Hu.

General results on precise asymptotics under sub-linear expectations

In this paper, we investigate the precise asymptotics for complete convergence and complete integral convergence under sub-linear expectations. Some general results on precise asymptotics are obtained, which improve and extend some existing ones in classical probability space and sub-linear expectation space. As applications, we further establish some general results on complete integral convergence under sub-linear expectations.

Central limit theorem for linear processes generated by m-dependent random variables under the sub-linear expectation

We prove the Rosnethal’s inequality of m-dependent random variables under the sub-linear expectation in this paper. Furthermore, we use this inequality to investigate the central limit theorem for linear processes generated by m-dependent random variables under sub-linear expectations. This article use the basic definitions of sub-linear expectation space, Kronecker lemma, Cr inequality etc. to demonstrate the main conclusion.

On the Hartman–Wintner law of the iterated logarithm under sublinear expectation

In this article, we establish two types of the Hartman–Wintner law of the iterated logarithm under sublinear expectations, especially for pseudo-independent random variables with the finite quadratic Choquet expectations, which generalizes the existing results for independent and identically distributed sequences. But unlike the classical situation, one counterexample is provided to show that the Hartman–Wintner law of the iterated logarithm may fail on the sublinear expectation space only with the finite second moment condition.

A complete convergence theorem of the maximum of partial sums under the sub-linear expectations

Let {X,Xn; n ? 0} be a sequence of independent and identically distributed random variables in a sub-linear expectation space (?,H,?). We establish a complete convergence theorem of the maximum of partial sums max1?j?n |?j i=1 Xi| under optimal moment condition in a sub-linear expectation space. Our result generalizes and improves the corresponding results.

A note on the cluster set of the law of the iterated logarithm under sub-linear expectations

In this note, we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space. For showing the result, a self-normalized law of the iterated logarithm is established.

Moderate deviation principle for $ m $-dependent random variables under the sub-linear expectation

<abstract><p>Let $ \{X_n, n\geq1\} $ be a sequence of $ m $-dependent strictly stationary random variables in a sub-linear expectation $ (\Omega, \mathcal{H}, \mathbb{E}) $. In this article, we give the definition of $ m $-dependent sequence of random variables under sub-linear expectation spaces taking values in $ \mathbb{R} $. Then we establish moderate deviation principle for this kind of sequence which is strictly stationary. The results in this paper generalize the result that in the case of independent identically distributed samples. It provides a basis to discuss the moderate deviation principle for other types of dependent sequences.</p></abstract>

The Moments of the Maximum of Normalized Partial Sums Related to Laws of the Iterated Logarithm Under the Sub-Linear Expectation∗

Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables on a sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, $S_n=X_1+\ldots+X_n$. We consider the moments of $\max_{n\ge 1}|S_n|/\sqrt{2n\log\log n}$. The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.

Complete convergence and complete integral convergence for weighted sums of widely acceptable random variables under the sub-linear expectations

<abstract> <p>Since the concept of sub-linear expectation space was put forward, it has well supplemented the deficiency of the theoretical part of probability space. In this paper, we establish the complete convergence and complete integration convergence for weighted sums of widely acceptable (abbreviated as WA) random variables under the sub-linear expectations with the different conditions. We extend the complete moment convergence in probability space to sublinear expectation space.</p> </abstract>

Complete convergence and complete integral convergence of partial sums for moving average process under sub-linear expectations

<abstract><p>In this paper, we establish the complete convergence and complete integral convergence of partial sums for moving average process based on independent random variables under the sub-linear expectations. The results in the paper extend some convergence properties of moving average process under independent assumption from probability space to the sub-linear expectation space.</p></abstract>

A general form for precise asymptotics for complete convergence under sublinear expectation

<abstract><p>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 <sup>[<xref ref-type="bibr" rid="b34">34</xref>]</sup>.</p></abstract>

Large deviation principle for linear processes generated by real stationary sequences under the sub-linear expectation

Let be the linear processes, where is an absolutely summable sequence of real numbers and is a sequence of real stationary independent random variables under the sub-linear expectation space In the framework of sub-linear expectation space, the Beveridge–Nelson decomposition of linear processes and the sub-additivity of capacity are used to establish the large deviation principle for linear processes in the sense of upper probability.

Strong limit theorems of weighted sums for extended negatively dependent random variables under sub-linear expectations

Limit theorems for sub-linear expectations are challenging field which have raised a large number of issues of interest recently. The aim of this article is to establish the strong limit theorems of weighted sums under a sub-linear expectation space. As applications, the strong limit theorem for extended negatively dependent and identically distributed random variables have been generalized to the sub-linear expectation space context.

The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation.

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.