Strong Law Research Articles

Degrees in random m-ary hooking networks

The theme in this paper is a composition of random graphs and Pólya urns. The random graphs are generated through a small structure called the seed. Via Pólya urns, we study the asymptotic degree structure in a random m-ary hooking network and identify strong laws. We further upgrade the result to second-order asymptotics in the form of multivariate Gaussian limit laws. We give a few concrete examples and explore some properties with a full representation of the Gaussian limit in each case. The asymptotic covariance matrix associated with the Pólya urn is obtained by a new method that originated in this paper and is reported in \cite{Mahmoud2022}.

Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions

ABSTRACT In this work, suppose that {X n ; n ≥ 1}is a sequence of asymptotically negatively associated random variables and {a ni ; 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers such that ∑ i = 1 n | a n i | q = O ( n ) for some q > max { α p − 1 α − 1 / 2 , 2 } with αp > 1 and α > 1 2 . Let l (x) > 0 be a slowly varying function at infinity. We establish some equivalent conditions of the complete convergence for weighted sums of this form ∑ n = 1 ∞ n α p − 2 l ( n ) P ( max 1 ≤ j ≤ n | ∑ i = 1 j a n i X i | > ε n α ) < ∞ for all ε > 0 . As applications, some strong laws of large numbers for weighted sums of asymptotically negatively associated random variables are also obtained.

Further results on laws of large numbers for the array of random variables under sub-linear expectation

– Motivated by risk measure, super-hedge pricing, and modeling uncertainty in finance, Shige Peng established the theory of sub-linear expectation. In this article, we derive two results of laws of large numbers in the framework of sub-linear expectations. One is the strong law of large numbers for the array of random variables, which satisfies non identical distributed and exponential negatively dependent under sub-linear expectation. The other is the weak law of large numbers for the array of random variables, which satisfies non identical distributed and Φ -negatively dependent under sub-linear expectation. These results include and extend some existing results.

Law of large numbers for weak Fubini-independent random variables under capacities without sub-additivity

This article attempts to establish the law of large numbers for random variables in Choquet expectation spaces where the Choquet integral is not sub-additive. As a basis, the Fubini-independence of random variables is generalized to a weaker version. Then the strong and weak law of large numbers are proved for weak-Fubini independent random variables under capacities which are not sub-additive.

Strong Laws of Large Numbers for Weighted Sums of Hilbert-valued Coordinatewise PNQD Random Variables with Applications

Strong Laws of Large Numbers for Weighted Sums of Hilbert-valued Coordinatewise PNQD Random Variables with Applications

Regime legitimation strategies and competition laws in autocracies

Competition laws play important roles in safeguarding and promoting competition in an economy. One important question is why some countries adopt strong competition laws, while others do not. Previous studies emphasize the difference between democracy and autocracy and argue that democracies tend to adopt stronger competition laws. However, these studies pay little attention to the fact that autocratic competition policies vary considerably. I argue that autocratic regimes’ legitimation strategies affect their competition policies. Even autocrats care about what ordinary citizens think and will attempt to legitimize themselves. However, autocrats rely on different legitimacy claims to justify their rule. When autocracies justify their rule based on their performance, they more seriously consider economic concentration and anti-competitive practices, which leads them to adopt stronger competition laws. This incentive is much weaker when they rely on other legitimacy claims. I further posit that institutional and external contexts modify the effect of performance-based legitimation strategies. To the extent that performance-based legitimation strategies lead to stronger competition policies, they are more likely to do so either when an authoritarian regime allows electoral competition or when neighboring countries have strong competition laws. I test the hypotheses by using a global sample of autocracies from 1950 to 2010. A comprehensive set of analyses provide strong evidence for the arguments. The result shows that an authoritarian regime’s performance-based legitimacy claims are positively related to its competition laws’ strength, while other legitimation strategies do not positively correlate with the strength of competition laws. Further analysis reveals that performance legitimacy claims significantly interact with electoral competition and the regional prevalence of competition laws to influence competition policies.

Cyclic Products and Optimal Traps in Cyclic Birth and Death Chains

A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.

Ideal management of social assistance funds as a prevention and mitigation effort corruption

Purpose: The purpose of this study is to determine the ideal legal arrangements for social assistance fund management to prevent corruption, and efforts to prevent and overcome corruption in social assistance funds. Research Methodology: The method used is normative juridical research using a statutory approach, a conceptual approach, and a sociological juridical approach. Results: The results showed that the ideal legal arrangement for the management of social assistance funds to prevent corruption is to apply some positive legal principles from John Austin such as being based on clear authorities such as legislative or government institutions, and the law must reflect state sovereignty, set clear rules and must be followed by all parties, Provide a foundation for strong and effective law enforcement, and the law can act as an instrument of control to prevent criminal acts of corruption. Efforts to prevent and overcome corruption of social assistance funds are through the application of Jeremy Bentham's theory of utilitarianism which has the main principle of maximizing overall social welfare or happiness, considering the distribution of benefits fairly, emphasizing transparency and accountability in decision making, and emphasizing transparency and accountability in decision making.

Invariant Measures for Uncountable Random Interval Homeomorphisms

A necessary and sufficient condition for the iterated function system {f(cdot ,omega ),|,omega in Omega } with probability P to have exactly one invariant measure mu _* with mu _*((0,1))=1 is given. The main novelty lies in the fact that we only require the transformations f(cdot ,omega ) to be increasing homeomorphims, without any smoothness condition, neither we impose conditions on the cardinality of Omega . In particular, positive Lyapunov exponents conditions are replaced with the existence of solutions to some functional inequalities. The stability and strong law of large numbers of the considered system are also proven.

Effects of Fe solid solute on grain boundaries of bi-crystal Cu: A molecular dynamics simulation

Grain boundaries (GBs) play a crucial role on the structural stability and mechanical properties of Cu and its alloys. In this work, molecular dynamics (MD) simulations are employed to study the effects of Fe solutes on the formation energy, excess volume, dislocations and melting behaviors of GBs in CuFe alloys. It is illustrated that Fe solute affects the structural stability of Cu GBs substantially, the formation energy of GBs is reduced, but the thickness and melting point of GBs are increased, that is, the structural stability of Cu GBs is significantly improved owing to the Fe solutes. A strong scaling law exists between the formation energy, excess volume, thickness and melting point of GBs. Therefore, Fe solid solute plays an important role in the characteristics of GBs in bi-crystal Cu.

The Kolmogorov strong law of large numbers for WOD random variables

The Kolmogorov strong law of large numbers for WOD random variables

Dynamic analysis of the IFCD rumor propagation model under stochastic disturbance on heterogeneous networks

By taking full consideration of the heterogeneity of network users and stochastic disturbance of network environment, a stochastic IFCD (Ignorants-Forwarders-Collectors-Deleters) rumor propagation model is proposed on heterogeneous networks. This study takes MicroBlog as the research background to investigate the dynamical behaviors of rumor spreading on online social networks, which paves the way for controlling rumors effectively and provides positive guidance for the network public opinion. Firstly, some preparations for building a stochastic model are given, including establishing a deterministic IFCD rumor propagation model and obtaining the rumor-free/spreading equilibrium. Secondly, considering the impact of stochastic disturbance on rumor propagation, the corresponding stochastic rumor propagation model is established. Thirdly, a stopping time is defined, which manifests the existence of the unique global positive solution in the stochastic model. Next, sufficient conditions of the rumor extinction and persistence are obtained by using Itô’s formula and strong law of large numbers. Additionally, simulations are operated to validate the results. Finally, the application of the model is shown to bear out the applicability of the proposed model.

Equivalent conditions of complete convergence and Marcinkiewicz–Zygmund-type strong law of large numbers for i.i.d. sequences under sub-linear expectations

Equivalent conditions of complete convergence and Marcinkiewicz–Zygmund-type strong law of large numbers for i.i.d. sequences under sub-linear expectations

Chow dilogarithm and strong Suslin reciprocity law

We prove a conjecture of A. Goncharov concerning strong Suslin reciprocity law. The main idea of the proof is the construction of the norm map on so-called lifted reciprocity maps. This construction is similar to the construction of the norm map on Milnor K K -theory. As an application, we express Chow dilogarithm in terms of Bloch-Wigner dilogarithm. Also, we obtain a new reciprocity law for four rational functions on an arbitrary algebraic surface with values in the pre-Bloch group.

Dynamic analysis and optimal control of a stochastic information spreading model considering super-spreader and implicit exposer with random parametric perturbations

Environmental factors in social systems affect information spreading at all times. This paper proposes a stochastic S2EIR model that considers the presence of super-spreaders and implicit exposers in information spreading, as well as the stochastic perturbation of model parameters. The existence of a global positive solution using the Itô′s formula is then demonstrated. Sufficient conditions for information disappearance and smooth distribution of information are calculated by using the Borel–Cantelli lemma and the strong law of large numbers. Furthermore, the optimal control strategy for the stochastic model is proposed using the Hamiltonian function. The results of the theoretical analysis are supported by numerical simulations and compared to the parameter variations of the deterministic model. The results of this study indicate that white noise facilitates the spread of information. The intensity of perturbation is proportional to the fluctuation of information spreading. Controlling random parameters can effectively facilitate the spread of information. For positive information, the randomness and complexity of the social system should be utilized to increase the spread of information. In contrast, for negative information, randomness in the social system should be suppressed to the greatest extent possible to limit information dissemination.

A monitoring and site visit intervention to reduce sales to minors at packaged liquor outlets.

Selling alcohol to a minor is a serious offence in the New South Wales Liquor Act. However, minors report they can purchase alcohol easily. This study used an intersectoral partnership between the Central Coast Local Health District (CCLHD) Health Promotion Service, and Liquor & Gaming NSW (L&GNSW) Compliance Operations to test a site visit intervention to increase ID checks at packaged liquor outlets (PLOs). As the current legislative framework does not allow minors to attempt to purchase alcohol from PLOs, even under compliance operations, this study employed pseudo-minors aged 18-19 years. Pseudo-minors visited all PLOs on the Central Coast, NSW in four survey rounds, attempting to purchase alcohol without ID. If a PLO sold alcohol to the pseudo-minor, they received a visit from a Health Promotion Officer and L&GNSW Inspector to inform them of the sale and that they are at risk of selling alcohol to a minor. Site visits were an effective intervention to increase the rates of ID checking. Alcohol sales to pseudo-minors without ID decreased from 34% in Round 1 to 7% in Round 4. Young sales staff and independent stores were less likely to check ID. This intervention was associated with some of the lowest rates of underage alcohol purchasing reported to date. However, it may not be sustainable in the long term. It is recommended that: future studies randomise site visits to investigate their contribution to the observed reduction, cost effectiveness of the intervention is explored and legislative change to allow controlled purchase operations for alcohol is considered. SO WHAT?: Young people under 18 years of age should not be able to purchase alcohol from liquor licence outlets, and strong laws apply if a sale is detected. However, despite these rules, young people still report they can access alcohol in this manner. Whilst our intervention of site visits with the regulatory body saw decreases in sales to our pseudo-minors, we advocate for controlled purchase operations (similar to tobacco control) as a more sustainable and effective intervention to reduce sales to minors - while this is not yet legislated for packaged liquor outlets in Australia, it is deemed best practice in other countries.

Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space

Abstract In this work, the authors study some convergence results including weak law of large numbers, strong law of large numbers, complete convergence, and complete moment convergence for weighted sums of coordinatewise asymptotically negatively associated random vectors in Hilbert spaces. These results improve or extend some corresponding ones in the literature.

Probabilistic Properties of Zipf Sets and Their Maximal Intersections

In the article, the weak and strong laws of large numbers are obtained for various characteristics of a generalized Zipf set. Based on these results, we investigate the maximal intersection size between a random Zipf set and the elements of a large base of independent random sets of the same type but, eventually, with different parameters. This question is significant for applications.

Role of Media in Coverage and Reporting of Child Abuse Cases

Child abuse is arguably one of the most debated publics/social policy areas. Thus, the need for systematic data is not only for recognizing child protection issues but also for formulating and implementing policy responses and practices in Pakistan. This study aims to analyze the role of the media in the research and reporting of child abuse cases. The method used in this study is to use media framing theory to conduct quantitative investigations using content analysis. Media coverage of child abuse can help communities understand issues around child abuse and neglect and interact constructively with children's rights. Using media as a reporting, representation, and advocacy tool will increase public knowledge about perpetrators and aid in preventing child abuse. However, there are shortcomings in structures and institutions that can cause children suffering. It needs cooperation from government agencies and civil society to improve child protection from maltreatment and carelessness. Therefore, the role of the media as a component of a more extensive system designed to protect children is crucial. Child sexual abuse can be reduced in the country by enacting strong anti-child sexual abuse laws. By affirming the country's commitment, providing legal protection, increasing public awareness, increasing the effectiveness of law enforcement. With clear laws in place, police and the justice system can act on child sexual abuse cases more quickly and decisively. This can provide a deterrent effect to perpetrators and prevent similar crimes in the future.

The Digital Personal Data Protection Bill 2022 in Contrast with the EU General Data Protection Regulation: A Comparative Analysis

The European Union’s General Data Protection Regulation (GDPR) is considered to be the most comprehensive & strong privacy and data protection law in the world, which doesn’t only regulate within the territory of EU but also has an extraterritorial effect. GDPR has influenced privacy & data protection legislation of many nations. India is ready with the draft Digital Personal Data Protection Bill, 2022 (DPDP Bill) which is the latest in a series of draft legislations presented and removed since mid-2018. In this article we discuss the key differences between the GDPR & DPDP Bill by analysing the different approaches and methods prescribed in both the legislations to understand their scope & applicability, concerned parties, classification of personal data, legal basis for data processing, children’s rights, reporting breach, cross-border data transfer, penalties, etc. In conclusion, we can say that the GDPR is relatively more detailed in its instructions, whereas the DPDP Bill establishes certain fundamental concepts. The DPDP Bill offers a glimpse of hope for balancing the interests of data subjects while acknowledging the practical challenges that businesses may encounter.