Sublinear Expectation Research Articles

Complete convergence and complete integral convergence for randomly weighted sums under the sublinear expectations

The complete convergence and complete integral convergence for randomly weighted sums when the weight is A n i under the sublinear expectations are established in this article. The major findings of this study are expansions of complete convergence and complete moment convergence for widely orthant-dependent (WOD) random variables in the traditional probability space.

Representation of weakly maxitive monetary risk measures and their rate functions

This article provides a representation result for monetary risk measures (i.e., monotone translation-invariant functionals) satisfying a weak maxitivity property. This result can be understood as a functional analytic generalization of the Gärtner-Ellis large deviations theorem. In contrast to the classical Gärtner-Ellis theorem, the rate function is computed on an arbitrary set of continuous real-valued functions rather than the dual space. As an application of the main result, we establish a large deviations result for sequences of sublinear expectations on regular Hausdorff topological spaces.

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.

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

<abstract><p>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.</p></abstract>

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.

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

<abstract><p>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<sup>[<xref ref-type="bibr" rid="b13">13</xref>]</sup>. We further extend the limit theorems in classical probability space.</p></abstract>

Complete and complete integral convergence for arrays of rowwise extended negatively dependent random variables under sublinear expectations

В статье изучается полная и полная интегральная сходимость для массивов построчно обобщенных отрицательно зависимых случайных величин относительно сублинейных ожиданий. Результаты настоящей статьи распространяют результат о полной сходимости, полученный Ху, Ваном и Розальским в 2015 г., и результат о полной моментной сходимости, полученный Ву, Ордоньесом Кабрерой и Володиным в 2014 г., с классического вероятностного пространства на пространство с сублинейным ожиданием.

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.

Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations

<abstract><p>In this article, we study the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations. We also give some sufficient assumptions for the convergence. Moreover, we get the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables. The results obtained in this paper generalize the relevant ones in probability space.</p></abstract>

A strong law of large numbers under sublinear expectations

A strong law of large numbers under sublinear expectations

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.

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 of moving average processes produced by negatively dependent random variables under sub-linear expectations

<abstract><p>Suppose that $ \{a_i, -\infty < i < \infty\} $ is an absolutely summable set of real numbers, $ \{Y_i, -\infty < i < \infty\} $ is a subset of identically distributed, negatively dependent random variables under sub-linear expectations. Here, we get complete convergence and Marcinkiewicz-Zygmund strong law of large numbers for the partial sums of moving average processes $ \{X_n = \sum_{i = -\infty}^{\infty}a_{i}Y_{i+n}, n\ge 1\} $ produced by $ \{Y_i, -\infty < i < \infty\} $ of identically distributed, negatively dependent random variables under sub-linear expectations, complementing the relevant results in probability space.</p></abstract>

The convergence rate for the laws of logarithms under sub-linear expectations

<abstract><p>Let $ \{X_n; n\geq1\} $ be a sequence of independent and identically distributed random variables in a sub-linear expectation space $ (\Omega, \mathcal{H}, \hat{\mathbb{E}}) $. The necessary and sufficient conditions for the convergence rate on the laws of the logarithms and the law of the iterated logarithm are obtained.</p></abstract>

A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation

This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for $ \alpha \in(1,2) $, the i.i.d. sequence $ \left\{ {\left( {\dfrac{1}{{\sqrt n }} \displaystyle\sum\limits_{i = 1}^n {{X_i}} ,\dfrac{1}{n} \displaystyle\sum\limits_{i = 1}^n {{Y_i}} ,\dfrac{1}{{\sqrt[\alpha ]{n}}} \displaystyle\sum\limits_{i = 1}^n {{Z_i}} } \right)} \right\}_{n = 1}^\infty $converges in distribution to $ \tilde{L}_{1} $, where $ \tilde{L}_{t}=(\tilde {\xi}_{t},\tilde{\eta}_{t},\tilde{\zeta}_{t}) $, $ t\in \lbrack0,1] $, is a multidimensional nonlinear Lévy process with an uncertainty set $ \Theta $ as a set of Lévy triplets. This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE) $\begin{aligned}[b]\left \{ \begin{array} {l} \partial_{t}u(t,x,y,z)-\sup \limits_{(F_{\mu},q,Q)\in \Theta }\left \{ \displaystyle\int_{\mathbb{R}^{d}}\delta_{\lambda}u(t,x,y,z)F_{\mu} ({\rm{d}}\lambda)\right. \\ \qquad\left. +\langle D_{y}u(t,x,y,z),q\rangle+\dfrac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right \} =0,\\ u(0,x,y,z)=\phi(x,y,z),\quad \forall(t,x,y,z)\in \lbrack 0,1]\times \mathbb{R}^{3d}, \end{array} \right.\end{aligned}$with $ \delta_{\lambda}u(t,x,y,z):=u(t,x,y,z+\lambda)-u(t,x,y,z)-\langle D_{z}u(t,x,y,z),\lambda \rangle $. To construct the limit process $ (\tilde {L}_{t})_{t\in \lbrack0,1]} $, we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space. We further prove a new type of Lévy-Khintchine representation formula to characterize $ (\tilde{L}_{t})_{t\in \lbrack0,1]} $. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.

Complete convergence and complete integration convergence for weighted sums of arrays of rowwise $ m $-END under sub-linear expectations space

<abstract><p>In this paper, we study the complete convergence and the complete integration convergence for weighted sums of $ m $-extended negatively dependent ($ m $-END) random variables under sub-linear expectations space with the condition of $ \hat{\mathbb{E}}|X|^p\leqslant C_{\mathbb{V}}(|X|^p) < \infty $, $ p > 1/\alpha $ and $ \alpha > 3/2 $. We obtain the results that can be regarded as the extensions of complete convergence and complete moment convergence under classical probability space. In addition, the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of $ m $-END random variables under the sub-linear expectations space is proved.</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).

Generalized Feynman–Kac formula under volatility uncertainty

In this paper we provide a generalization of a Feynmac–Kac formula under volatility uncertainty in presence of a linear term in the PDE due to discounting. We state our result under different hypothesis with respect to the derivation given by Hu et al. (2014), where the Lipschitz continuity of some functionals is assumed which is not necessarily satisfied in our setting. In particular, we show that the G-conditional expectation of a discounted payoff is a viscosity solution of a nonlinear PDE. In applications, this permits to calculate such a sublinear expectation in a computationally efficient way.

Imbalanced binary classification under distribution uncertainty

Imbalanced binary classification plays an important role in many applications. Some popular classifiers, such as logistic regression (LR), usually underestimate the probability of the minority class. Therefore, in this paper, we introduce two novel methods under distribution uncertainty, the idea of which is to modify the predicted probability with an additional uncertainty estimation. We develop the mean-uncertain method and the volatility-uncertain method, respectively, by assuming that the disturbance term follows the maximal and the G-normal distributions, which are the most important distributions within a sublinear expectation framework. Experiments on the simulated dataset and 10real-life datasets are conducted to compare the newly proposed approaches to several existing ones, including two resampling methods and two regression-based methods. The results of experiments show that our methods outperform most of the others in common evaluation metrics, especially the accuracy of the minority class.

Stability of stochastic Hopfield neural networks driven by [formula omitted]-Brownian motion with time-varying and distributed delays

In this paper, pth exponential stability and quasi-surely exponential stability of stochastic Hopfield neural networks driven by G-Brownian motion are investigated. Under a sublinear expectation framework, we give several lemmas related to the Halanay inequality and the Burkholder-Davis-Gundy ineqaulity. Our stability results is based on the class of the multidimensional Halanay inequalities, as well as the Burkholder-Davis-Gundy inequalities. The main originality lies in the fact that we consider Knightian uncertainty of the theory model of stochastic Hopfield neural networks. Finally, two numerical examples are presented to illustrate our new theory.