Sublinear Expectation Space Research Articles

The convergence properties for randomly weighted sums of widely negative dependent random variables under sub-linear expectations with related statistical applications

In this paper, we study the complete convergence and complete moment convergence for randomly weighted sums of arrays of rowwise widely negative dependent random variables in sub-linear expectation space under some appropriate conditions, which extend and improve the corresponding ones in classical probability space to the case of sub-linear expectation space. And we obtain a strong law of large numbers for the randomly weighted sums of arrays of rowwise widely negative dependent random variables. As applications of our main results, we not only present a result on the complete consistency for the weighted estimator in a nonparametric regression model, but also obtain the complete consistency for the least squares estimators in errors-in-variables regression models based on widely negative dependent errors under sub-linear expectations. We perform some numerical simulations to verify the validity of the theoretical results and a real example is analysed for illustration.

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

. This article aims to study and establish the Lai law for the weighted sum of extended negatively dependent random variables under sub-linear expectations. Using inequalities under sub-linear expectation spaces, we extend the complete convergence of weighted sums from classical probability space to sub-linear expectation space.

A monotone scheme for G-equations with application to the explicit convergence rate of robust central limit theorem

We propose a monotone approximation scheme for a class of fully nonlinear PDEs called G-equations. Such equations arise often in the characterization of G-distributed random variables in a sublinear expectation space. The proposed scheme is constructed recursively based on a piecewise constant approximation of the viscosity solution to the G-equation. We establish the convergence of the scheme and determine the convergence rate with an explicit error bound, using the comparison principles for both the scheme and the equation together with a mollification procedure. The first application is obtaining the convergence rate of Peng's robust central limit theorem with an explicit bound of Berry-Esseen type. The second application is an approximation scheme with its convergence rate for the Black-Scholes-Barenblatt equation.

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

. In this article, we study and establish complete convergence for widely negative orthant dependent random variables under the sub-linear expectations, which extend some complete convergence theorems for widely negative orthant dependent random variables from the classical probability space to sub-linear expectation space.

Strong law of large numbers for weighted sums of $ m $-widely acceptable random variables under sub-linear expectation space

<p>In this article, using the Fuk-Nagaev type inequality, we studied general strong law of large numbers for weighted sums of $ m $-widely acceptable ($ m $-WA, for short) random variables under sublinear expectation space with the integral condition</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \hat{\mathbb{E}} \left ( f^-\left ( \left | X \right | \right ) \right ) \le \mathrm{C}_\mathbb{V}\left ( f^-\left ( \left | X \right | \right ) \right )< \infty $\end{document} </tex-math></disp-formula></p><p>and $ Choquet $ integrals existence, respectively, where</p><p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ f\left ( x \right ) = x^{1/\beta }L\left ( x \right ) $\end{document} </tex-math></disp-formula></p><p>for $ \beta > 1 $, $ L\left (x \right) > 0 $ $ \left(x > 0\right) $ was a monotonic nondecreasing slowly varying function, and $ f^-\left (x \right) $ was the inverse function of $ f\left(x\right) $. One of the results included the Kolmogorov-type strong law of large numbers and the partial Marcinkiewicz-type strong law of large numbers for $ m $-WA random variables under sublinear expectation space. Besides, we obtained almost surely convergence for weighted sums of $ m $-WA random variables under sublinear expectation space. These results improved the corresponding results of Ma and Wu under sublinear expectation space.</p>

Almost sure convergence for a class of dependent random variables under sub-linear expectations

<abstract><p>This article aimed to investigate the almost sure convergence theorem of widely negative orthant dependent (WNOD) random variables under sub-linear expectation space. The conclusions in this essay are an extension of the corresponding conclusions in the classical probability space.</p></abstract>

On the complete moment convergence of moving average processes generated by negatively dependent random variables under sub-linear expectations

<abstract><p>The moving average processes $ X_k = \sum_{i = -\infty}^{\infty}a_{i+k}Y_{i} $ are studied, where $ \{Y_i, -\infty < i < \infty\} $ is a double infinite sequence of negatively dependent random variables under sub-linear expectations, and $ \{a_i, -\infty < i < \infty\} $ is an absolutely summable sequence of real numbers. We establish the complete moment convergence of a moving average process under proper conditions, extending the corresponding results in classic probability space to those in sub-linear expectation space.</p></abstract>

Some convergence properties for arrays of rowwise asymptotically almost negatively associated random variables under sub-linear expectations

. In this article, under some suitable conditions, we study the L q convergence and complete q-th moment convergence for arrays of rowwise asymptotically almost negatively associated (AANA, for short) random variables under sub-linear expectations. Some general results on L q convergence and complete q-th moment convergence for arrays of rowwise AANA random variables under sub-linear expectations are established, which extend the corresponding ones from classic probability space to the case of sub-linear expectation space. As applications of our main results, we not only present a result on the complete consistency of the weighted estimator in a nonparametric regression model but also obtain the complete consistency of the estimators in errors-in-variables (EV, for short) regression models based on AANA errors under sub-linear expectations. We perform some numerical simulations to verify the validity of the theoretical results.

Complete convergence for moving average process generated by extended negatively dependent random variables under sub-linear expectations

. In this article, the complete convergence for the partial sum of the moving average process { X n = ∑ i = − ∞ ∞ a i Y i + n , n ≥ 1 } is established under some mild conditions, where { Y i , − ∞ < i < ∞ } is a sequence of extended negatively dependent and identically distributed random variables that is stochastically dominated by a random variable Y under a sub-linear expectation ( Ω , H , E ̂ ) , and { a i , − ∞ < i < ∞ } is an absolutely summable sequence of real numbers. These conclusions promote and improve the corresponding results of moving the average process under extended negatively dependent random variables from the traditional probability space to the sub-linear expectation space.

Further research on complete integral convergence for moving average process of ND random variables under sub-linear expectations

In this article, we establish some results on complete convergence and complete integral convergence of moving average process { X n = ∑ i = − ∞ ∞ a i Y i + n , n ≥ 1 } based on negatively dependent random variables under sub-linear expectations. We generalize corresponding conclusions obtained by Zhang and Ding, and extend convergence theorems for general function from classical probability space to the sub-linear expectation space.

Equivalent Conditions of Complete p-th Moment Convergence for Weighted Sum of ND Random Variables under Sublinear Expectation Space

We investigate the complete convergence for weighted sums of sequences of negative dependence (ND) random variables and p-th moment convergence for weighted sums of sequences of ND random variables under sublinear expectation space. Using moment inequality and truncation methods, we prove the equivalent conditions of complete convergence for weighted sums of sequences of ND random variables and p-th moment convergence for weighted sums of sequences of ND random variables under sublinear expectation space.

Moderate Deviation Principle for Linear Processes Generated by Dependent Sequences under Sub-Linear Expectation

We are interested in the linear processes generated by dependent sequences under sub-linear expectation. Using the Beveridge–Nelson decomposition of linear processes and the inequalities, the moderate deviation principle for linear processes produced by an m-dependent sequence is established. We also prove the upper bound of the moderate deviation principle for linear processes produced by negatively dependent sequences via different methods from m-dependent sequences. These conclusions promote and improve the corresponding results from the traditional probability space to the sub-linear expectation space.

Chover’s law of the iterated logarithm for weighted sums under sub-linear expectations

In this article, the authors prove the Chover-type law of the iterated logarithm for weighted partial sums of independent and identically distributed random variables, and also give some classical examples. These conclusions promote and improve the corresponding results from the traditional probability space to the sub-linear expectation space.

Complete convergence theorems for moving average process generated by independent random variables under sub-linear expectations

The research of convergence properties of moving average process is a challenging field of limit theorems. The aim of this article is to provide a method to prove the complete convergence and complete integral convergence of moving average process for independent random variables in sub-linear expectation space. The results obtained in the article are the extensions of some complete convergence theorems under classical probability space.

Capacity inequalities and strong laws for m-widely acceptable random variables under sub-linear expectations

In this work, a generalised concept of dependent random variables, i.e., m-widely acceptable random variables, is put forward and some inequalities such as exponential inequality and Rosenthal-type inequality are established under sub-linear expectations. By virtue of these inequalities, we obtain a general result on complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of m-widely acceptable random variables, which extend and improve some existing ones in classical probability space as well as in sub-linear expectation space. As applications of the main results, the complete consistency and strong consistency of weighted estimators in nonparametric regression model under sub-linear expectations are also investigated.

Complete convergence and complete moment convergence for widely negative orthant dependent random variables under the sub-linear expectations

In this work there is considered complete convergence and complete moment convergence for widely negative orthant dependent random variables under the sub-linear expectations. The presented results concern the weighted sums of these random variables and extend the corresponding results in classical probability space to the case of sub-linear expectation space.

Precise asymptotics for complete integral convergence in the law of the logarithm under the sub-linear expectations

&lt;abstract&gt;&lt;p&gt;The aim of this paper is to study and establish precise asymptotics for complete integral convergence in the law of the logarithm under the sub-linear expectation space. The methods and tools in this paper are different from those used to study precise asymptotics theorems in probability space. We extend precise asymptotics for complete integral convergence from the classical probability space to sub-linear expectation space. Our results generalize corresponding results obtained by Fu and Yang&lt;sup&gt;[&lt;xref ref-type="bibr" rid="b13"&gt;13&lt;/xref&gt;]&lt;/sup&gt;. We further extend the limit theorems in classical probability space.&lt;/p&gt;&lt;/abstract&gt;

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

In the paper, the complete convergence for weighted sums of arrays of rowwise widely negative dependent random variables under the sub-linear expectations is established. The main result partially extend some known results both in the classical probability space and sub-linear expectation space. In addition, it weaken their conditions.

Convergence of series and almost sure convergence for weighted random variables under sub-linear expectations

In this paper, convergence of series and almost sure convergence are established for weighted random variables under a sub-linear expectation space. Our results are very extensive versions which contain the related convergence of series and almost sure convergence for sequences of random variables and so on, and are extensions and improvements of classical convergence of series and almost sure convergence from the traditional probability space to the sub-linear expectation space.

Exponential inequalities and a strong law of large numbers for END random variables under sub-linear expectations

&lt;abstract&gt;&lt;p&gt;The focus of our work is to investigate exponential inequalities for extended negatively dependent (END) random variables in sub-linear expectations. Through these exponential inequalities, we were able to establish the strong law of large numbers with convergence rate $ O\left(n^{-1/2}\ln^{1/2}n\right) $. Our findings in sub-linear expectation spaces have extended the corresponding results previously established in probability space.&lt;/p&gt;&lt;/abstract&gt;