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## A Berry--Esseen theorem for weakly negatively dependent random variables and its applications

Acta Mathematica Hungarica
Authors:
Jianfeng Wang
and
Lixin Zhang

## Summary

We provide uniform rates of convergence in the central limit theorem for linear negative quadrant dependent (LNQD) random variables. Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\{X_{n},\allowbreak n\ge1\}$ \end{document} be a LNQD sequence of random variables with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $EX_{n}=0$ \end{document} , set \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S_{n}=\sum_{j=1}^{n}X_{j}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $B_{n}^{2}=\text{Var}\, (S_{n})$ \end{document} . We show that \begin{gather*} \sup_{x} \left|P\left(\frac{S_{n}}{B_{n}}<x\right)-\Phi(x)\right|= O\bigg(n^{-\delta/(2+3\delta)}\vee \frac{n^{3\delta^{2}/(4+6\delta)}}{B^{2+\delta}_{n}} \sum_{i=1}^{n} E{|X_{i}|}^{2+\delta}\bigg) \end{gather*} under finite \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(2+\delta)$ \end{document} th moment and a power decay rate of covariances. Moreover, by the truncation method, we obtain a Berry--Esseen type estimate for negatively associated (NA) random variables with only finite second moment. As applications, we obtain another convergence rate result in the central limit theorem and precise asymptotics in the law of the iterated logarithm for NA sequences, and also for LNQD sequences.

Restricted access

Authors:
Jianfeng Wang
and
Hong Li

## Abstract

### Background and aims

A growing body of evidence indicates a connection between emotional processes and the emergence and progression of addiction. However, scant research has examined the involvement of emotional processing within the framework of problematic pornography use (PPU). This study aimed to examine the electrophysiological and subjective differences in emotional processing between male individuals with PPU and healthy controls (HCs) following exposure to everyday affective images.

### Methods

Event-related potentials (ERPs) were recorded from 42 PPU participants (mean age = 20.14 years, SD = 1.35) and 45 HCs (mean age = 20.04 years, SD = 1.45) during an oddball task, in which unpleasant, pleasant, and neutral images were presented as deviant stimuli, while a neutral kettle image served as the standard stimulus. The Self-Assessment Manikin (SAM) was employed to assess participants' subjective experience on the dimensions of valence and arousal.

### Results

Regarding subjective measures of emotion, individuals with PPU reported lower valence ratings for unpleasant images compared to HCs. In terms of electrophysiological measures of emotion, PPU participants reported larger P2 amplitudes for unpleasant pictures compared to both pleasant and neutral pictures. Moreover, HCs showed enhanced P3 amplitudes in response to pleasant images compared to neutral images, whereas this effect was not observed in PPU participants.

### Discussion and Conclusion

These findings indicate that individuals with PPU may display deficits in emotional processing characterized by enhanced responsiveness to negative stimuli and attenuated responsiveness to positive stimuli. The heightened sensitivity to negative stimuli may contribute to the inclination of individuals with PPU to engage in pornography as a coping mechanism for stress regulation. Conversely, their diminished sensitivity to positive stimuli presents a challenge in seeking alternative natural rewards to counter potentially addictive behaviors.

Open access

Authors:
Jianfeng Wang
and
Bing Dai

## Abstract

### Background and aims

Impaired behavioral inhibitory control (BIC) is known to play a crucial role in addictive behavior. However, research has been inconclusive as to whether this is also the case for cybersex addiction. This study aimed to investigate the time course of BIC in male individuals with tendencies towards cybersex addiction (TCA) using event-related potentials (ERPs) and to provide neurophysiological evidence of their deficient BIC.

### Methods

Thirty-six individuals with TCA and 36 healthy controls (HCs) were given a Two-Choice Oddball task that required them to respond differently to frequent standard stimuli (images of people) and infrequent deviant stimuli (pornographic images) within 1,000 ms. Electroencephalography (EEG) was recorded as the participants performed the task.

### Results

Despite the similarity of standard stimuli between the groups in terms of reaction times (RTs), the RTs of the TCA group to deviant stimuli were much slower than those of the HC group. The behavioral difference was accompanied by group differences in the averaged amplitudes of N2 (200–300 ms) and P3 (300–500 ms) components in the deviant-standard difference wave. More specifically, compared to the HC group, the TCA group demonstrated smaller N2 and P3 amplitude differences for deviant than standard stimuli.

### Discussion and conclusions

Individuals with TCA were more impulsive than HC participants and shared neuropsychological and ERP characteristics of substance use disorder or behavioral addictions, which supports the view that cybersex addiction can be conceptualized as a behavioral addiction.

Open access

Authors:
Jianfeng Wang
,
Yuanyuan Chen
, and
Hui Zhang

## Abstract

### Background and aims

Attentional bias is a key factor in addictive behavior maintenance. However, whether attentional bias has a similar effect on cybersex addiction is unclear. We investigated differences in the attentional processing of sexually explicit images between individuals with high tendencies toward cybersex addiction (TCA) versus low tendencies using behavioral and electrophysiological indices.

### Methods

Twenty-eight individuals with high TCA and 29 with low TCA performed an addiction Stroop task comprising sexual and neutral images in colored frames. Participants were asked to respond to the frame color and not the image contents, and behavioral and event-related potentials were recorded.

### Results

Behaviorally, an addiction Stroop interference effect was found in the high TCA group, as shown by the longer reaction times to judge the frame colors of sexual images. Electrophysiologically, a P200 (150–220 ms) enhancement was present in response to sexual images compared with neutral ones, which was absent in the low TCA group. The event-related potential correlates with the addiction Stroop interference effect, indicating that the attentional bias underlying the addiction Stroop interference operates at an automatic level. A general, sexually related bias was found in the late positive potential (300–700 ms) amplitude, although between-group differences were insignificant.

### Discussion and conclusions

These findings indicate that sexual stimuli grab the attentional resources of individuals with high TCA at early automatic stages of attentional processing. Increased cue reactivity to sexual stimuli may contribute to pornographic consumption and play a crucial role in sustaining problematic excessive use of online pornography.

Open access