A remark on the weak convergence of processes in the Skorohod topology

In this paper,there are new considerations about the dual of a modular spaces and weak convergence. Two common fixed point theorems for a -non-expansive mapping defined on a star-shaped weakly compact subset are proved, Here the conditions of affineness, demi-closedness and Opial's property play an active role in the proving our results.

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Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficients

Stochastic Processes and their Applications ◽

10.1016/j.spa.2020.12.003 ◽

2021 ◽

Vol 134 ◽

pp. 55-93

Author(s):

Jianbo Cui ◽

Jialin Hong ◽

Liying Sun

Keyword(s):

Invariant Measure ◽

Weak Convergence ◽

Full Discretization

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Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

Journal of Applied Probability ◽

10.1017/jpr.2020.96 ◽

2021 ◽

Vol 58 (2) ◽

pp. 372-393

Author(s):

H. M. Jansen

Keyword(s):

Markov Chain ◽

Weak Convergence ◽

Regime Switching ◽

Sufficient Conditions ◽

The State ◽

Stochastic Integrals ◽

Occupation Measure ◽

Generator Matrix ◽

Markov Modulated ◽

Erlang Loss

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

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Weak Convergence in Poisson and Lévy Approximation Schemes

10.1002/9781119851257.ch2 ◽

2021 ◽

pp. 57-106

Keyword(s):

Weak Convergence ◽

Approximation Schemes

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A Mean Extragradient Method for Solving Variational Inequalities

Symmetry ◽

10.3390/sym13030462 ◽

2021 ◽

Vol 13 (3) ◽

pp. 462

Author(s):

Apichit Buakird ◽

Nimit Nimana ◽

Narin Petrot

Keyword(s):

Hilbert Space ◽

Variational Inequality ◽

Weak Convergence ◽

Numerical Experiments ◽

Variational Inequality Problem ◽

Extragradient Method ◽

Mean Value ◽

Inequality Problem ◽

The Mean ◽

Mean Value Method

We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann’s mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.

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Moderate deviation principle for multivalued stochastic differential equations

Stochastics and Dynamics ◽

10.1142/s021949372050015x ◽

2019 ◽

Vol 20 (03) ◽

pp. 2050015 ◽

Cited By ~ 1

Author(s):

Hua Zhang

Keyword(s):

Brownian Motion ◽

Weak Convergence ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Moderate Deviation ◽

Reflected Brownian Motion ◽

Moderate Deviation Principle ◽

Deviation Principle ◽

Weak Convergence Approach

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

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Asymptotic behavior of the records of multivariate random sequences in a norm sense

Mathematica Slovaca ◽

10.1515/ms-2017-0442 ◽

2020 ◽

Vol 70 (6) ◽

pp. 1457-1468

Author(s):

Haroon M. Barakat ◽

M. H. Harpy

Keyword(s):

Asymptotic Behavior ◽

Weak Convergence ◽

Sufficient Conditions ◽

Record Values ◽

Necessary And Sufficient Conditions ◽

Random Sequences ◽

Necessary And Sufficient

AbstractIn this paper, we investigate the asymptotic behavior of the multivariate record values by using the Reduced Ordering Principle (R-ordering). Necessary and sufficient conditions for weak convergence of the multivariate record values based on sup-norm are determined. Some illustrative examples are given.

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