In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theory, but the quantity in the role taken by the variance in the classic CLT has a more complicated definition. See also the general form of Bienaymé's identity. == Statement == Suppose that: the sequence X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } of random elements of some set is a Markov chain that has a stationary probability distribution; and the initial distribution of the process, i.e. the distribution of X 1 {\textstyle X_{1}} , is the stationary distribution, so that X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } are identically distributed. In the classic central limit theorem these random variables would be assumed to be independent, but here we have only the weaker assumption that the process has the Markov property; and g {\textstyle g} is some (measurable) real-valued function for which var ( g ( X 1 ) ) < + ∞ . {\textstyle \operatorname {var} (g(X_{1}))<+\infty .} Now let μ = E ( g ( X 1 ) ) , μ ^ n = 1 n ∑ k = 1 n g ( X k ) σ 2 := lim n → ∞ var ( n μ ^ n ) = lim n → ∞ n var ( μ ^ n ) = var ( g ( X 1 ) ) + 2 ∑ k = 1 ∞ cov ( g ( X 1 ) , g ( X 1 + k ) ) . {\displaystyle {\begin{aligned}\mu &=\operatorname {E} (g(X_{1})),\\{\widehat {\mu }}_{n}&={\frac {1}{n}}\sum _{k=1}^{n}g(X_{k})\\\sigma ^{2}&:=\lim _{n\to \infty }\operatorname {var} ({\sqrt {n}}{\widehat {\mu }}_{n})=\lim _{n\to \infty }n\operatorname {var} ({\widehat {\mu }}_{n})=\operatorname {var} (g(X_{1}))+2\sum _{k=1}^{\infty }\operatorname {cov} (g(X_{1}),g(X_{1+k})).\end{aligned}}} Then as n → ∞ , {\textstyle n\to \infty ,} we have n ( μ ^ n − μ ) → D Normal ( 0 , σ 2 ) , {\displaystyle {\sqrt {n}}({\hat {\mu }}_{n}-\mu )\ {\xrightarrow {\mathcal {D}}}\ {\text{Normal}}(0,\sigma ^{2}),} where the decorated arrow indicates convergence in distribution. == Monte Carlo Setting == The Markov chain central limit theorem can be guaranteed for functionals of general state space Markov chains under certain conditions. In particular, this can be done with a focus on Monte Carlo settings. An example of the application in a MCMC (Markov Chain Monte Carlo) setting is the following: Consider a simple hard spheres model on a grid. Suppose X = { 1 , … , n 1 } × { 1 , … , n 2 } ⊆ Z 2 {\displaystyle X=\{1,\ldots ,n_{1}\}\times \{1,\ldots ,n_{2}\}\subseteq Z^{2}} . A proper configuration on X {\displaystyle X} consists of coloring each point either black or white in such a way that no two adjacent points are white. Let χ {\displaystyle \chi } denote the set of all proper configurations on X {\displaystyle X} , N χ ( n 1 , n 2 ) {\displaystyle N_{\chi }(n_{1},n_{2})} be the total number of proper configurations and π be the uniform distribution on χ {\displaystyle \chi } so that each proper configuration is equally likely. Suppose our goal is to calculate the typical number of white points in a proper configuration; that is, if W ( x ) {\displaystyle W(x)} is the number of white points in x ∈ χ {\displaystyle x\in \chi } then we want the value of E π W = ∑ x ∈ χ W ( x ) N χ ( n 1 , n 2 ) {\displaystyle E_{\pi }W=\sum _{x\in \chi }{\frac {W(x)}{N_{\chi }{\bigl (}n_{1},n_{2}{\bigr )}}}} If n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} are even moderately large then we will have to resort to an approximation to E π W {\displaystyle E_{\pi }W} . Consider the following Markov chain on χ {\displaystyle \chi } . Fix p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} and set X 1 = x 1 {\displaystyle X_{1}=x_{1}} where x 1 ∈ χ {\displaystyle x_{1}\in \chi } is an arbitrary proper configuration. Randomly choose a point ( x , y ) ∈ X {\displaystyle (x,y)\in X} and independently draw U ∼ U n i f o r m ( 0 , 1 ) {\displaystyle U\sim \mathrm {Uniform} (0,1)} . If u ≤ p {\displaystyle u\leq p} and all of the adjacent points are black then color ( x , y ) {\displaystyle (x,y)} white leaving all other points alone. Otherwise, color ( x , y ) {\displaystyle (x,y)} black and leave all other points alone. Call the resulting configuration X 1 {\displaystyle X_{1}} . Continuing in this fashion yields a Harris ergodic Markov chain { X 1 , X 2 , X 3 , … } {\displaystyle \{X_{1},X_{2},X_{3},\ldots \}} having π {\displaystyle \pi } as its invariant distribution. It is now a simple matter to estimate E π W {\displaystyle E_{\pi }W} with w n ¯ = ∑ i = 1 n W ( X i ) / n {\displaystyle {\overline {w_{n}}}=\sum _{i=1}^{n}W(X_{i})/n} . Also, since χ {\displaystyle \chi } is finite (albeit potentially large) it is well known that X {\displaystyle X} will converge exponentially fast to π {\displaystyle \pi } which implies that a CLT holds for w n ¯ {\displaystyle {\overline {w_{n}}}} . == Implications == Not taking into account the additional terms in the variance which stem from correlations (e.g. serial correlations in markov chain monte carlo simulations) can result in the problem of pseudoreplication when computing e.g. the confidence intervals for the sample mean.
Inauthentic text
An inauthentic text is a computer-generated expository document meant to appear as genuine, but which is actually meaningless. Frequently they are created in order to be intermixed with genuine documents and thus manipulate the results of search engines, as with Spam blogs. They are also carried along in email in order to fool spam filters by giving the spam the superficial characteristics of legitimate text. Sometimes nonsensical documents are created with computer assistance for humorous effect, as with Dissociated press or Flarf poetry. They have also been used to challenge the veracity of a publication—MIT students submitted papers generated by a computer program called SCIgen to a conference, where they were initially accepted. This led the students to claim that the bar for submissions was too low. With the amount of computer generated text outpacing the ability of people to humans to curate it, there needs some means of distinguishing between the two. Yet automated approaches to determining absolutely whether a text is authentic or not face intrinsic challenges of semantics. Noam Chomsky coined the phrase "Colorless green ideas sleep furiously" giving an example of grammatically correct, but semantically incoherent sentence; some will point out that in certain contexts one could give this sentence (or any phrase) meaning. The first group to use the expression in this regard can be found below from Indiana University. Their work explains in detail an attempt to detect inauthentic texts and identify pernicious problems of inauthentic texts in cyberspace. The site has a means of submitting text that assesses, based on supervised learning, whether a corpus is inauthentic or not. Many users have submitted incorrect types of data and have correspondingly commented on the scores. This application is meant for a specific kind of data; therefore, submitting, say, an email, will not return a meaningful score.
WS-SecurityPolicy
WS-Security Policy is a web services specification, created by IBM and 12 co-authors, that has become an OASIS standard as of version 1.2. It extends the fundamental security protocols specified by the WS-Security, WS-Trust and WS-Secure Conversation by offering mechanisms to represent the capabilities and requirements of web services as policies. Security policy assertions are based on the WS-Policy framework. Policy assertions can be used to require more generic security attributes like transport layer security
Webull
Webull Corporation, often stylized as simply Webull, is a U.S.-based financial services holding company headquartered in St. Petersburg, Florida. It owns and operates the Webull electronic trading platform for self-directed retail investors. Depending on jurisdiction, the Webull platform offers trading in stocks, exchange-traded funds (ETFs), options, margin, bonds, cryptocurrency and futures, as well as market-data tools. Webull began operations in 2016 under Hunan Fumi Information Technology, a China-based financial technology company founded by Wang Anquan. It launched U.S. brokerage services through Webull Financial LLC in 2018 and expanded during the retail-trading boom of 2020 and 2021. In April 2025, Webull became a publicly traded company on the Nasdaq through a merger with special-purpose acquisition company SK Growth Opportunities Corporation. The company's U.S. brokerage revenue relies substantially on payment for order flow, with options trading accounting for the larger share of its order-flow rebates in 2025. Webull has faced regulatory actions related to options customer approvals, complaint handling, suspicious activity reporting, social-media marketing and customer disclosures. It has also faced scrutiny from U.S. lawmakers and state officials over its historical and operational ties to China and the handling of U.S. customer data. == History == === Founding === Webull was founded in 2016 under Hunan Fumi Information Technology, a China-based financial technology company, by Wang Anquan, a former employee of Alibaba Group and Xiaomi. Hunan Fumi Information Technology received backing from Xiaomi, Shunwei Capital, and other investors in China. Fumi Technology was a Hunan-based fintech start-up incubated by Xiaomi and raised about CNY200 million (approximately US$30 million) in a Series B financing round in 2018. On May 24, 2017, Webull Financial LLC was established as a Delaware limited liability company. It began offering brokerage services in the United States in May 2018. Wang hired Anthony Denier as CEO of the U.S. brokerage that year and the two mapped out their strategy on napkins at a Mexican restaurant in New York City. Webull Corporation was incorporated in the Cayman Islands in September 2019 as the group's holding company. === Retail trading boom === In May 2020, the company received SEC approval to launch a robo-advisor on its platform. By August 2020, the platform had over 11 million registered users, and in October 2020, it had 750,000 daily active users. Webull introduced options trading in 2020 and later added cryptocurrency trading through a separate digital-asset business. In November 2020, Webull began supporting cryptocurrency transactions. In December 2020, Webull launched trading services in Hong Kong. During the GameStop short squeeze in January 2021, Webull gained attention as some retail traders looked for alternatives to Robinhood. On January 27, 2021, Webull recorded its highest-ever number of active daily users, at 952,000, and the Webull app was downloaded across the Apple App and Google Play stores an estimated 100,000 times. That week, approximately 1.2 million people downloaded the Webull mobile app, which the company reported as a 1,548% week-over-week increase. On January 28, 2021, Webull was directed by its clearing house to temporarily halt buy orders for stocks affected by the GameStop short squeeze. In June 2021, Webull was reported to be considering a U.S. initial public offering that could raise up to $400 million. === Restructuring and expansion === Webull restructured its China-related corporate arrangements in 2022 and later stated that Hunan Fumi was no longer affiliated with the group. In 2022 and 2023, Webull expanded in several non-U.S. markets, including Singapore, Australia, South Africa, Japan, the United Kingdom and Indonesia. In June 2023, Webull moved cryptocurrency trading to a separate app called Webull Pay. By the end of 2023, Webull had 4.3 million funded accounts and US$8.2 billion in customer assets. In January 2024, Anthony Denier was promoted to group president of Webull Corporation. In November 2024, Webull launched overnight, or extended-hours, trading, expanding the trading window of U.S. stocks for users inside and outside the United States. === SPAC merger and Nasdaq listing === On February 28, 2024, Webull agreed to go public through a business combination with SK Growth Opportunities Corporation (NASDAQ: SKGR), a special-purpose acquisition company, in a deal that valued the company at approximately US$7.3 billion. The proposed valuation drew scrutiny because of Webull's limited financial disclosure at announcement, reliance on payment for order flow and small expected public float. SK Growth shareholders approved the business combination on March 30, 2025, and the transaction closed on April 10, 2025. Webull's Class A ordinary shares and warrants began trading on the Nasdaq on April 11, 2025 under the ticker symbols BULL and BULLW (incentive warrants traded under BULLZ until their redemption in June 2025). The merger brought Webull to the public market but generated little cash for the company: after shareholder redemptions, Webull disclosed net proceeds of US$430,066 from the transaction. After the listing, Webull's shares experienced extreme volatility, rising as much as 500% to US$79.56 on April 14, 2025, after closing at US$13.25 on the prior trading day. The initial post-listing surge increased the value of Webull holdings owned by earlier investors, including RIT Capital Partners, which had first invested in Webull in 2021. In April 2026, after Webull's shares had fallen about 70% over the previous year, the company authorized a US$100 million share repurchase program. == Business model and financials == Webull provides a self-directed electronic trading platform available through mobile, desktop and web applications. Depending on jurisdiction, the platform offers trading in stocks, exchange-traded funds, options, margin, futures, fixed income products, cryptocurrency, cash management features and market data tools. In the United States, Webull Financial LLC is a registered broker-dealer and member of FINRA and the Securities Investor Protection Corporation, while Webull operates in other markets through locally licensed brokerage subsidiaries. Webull operates a commission-free or low-cost brokerage model for self-directed retail investors. In the United States, a substantial part of its trading-related revenue comes from payment for order flow, while in some non-U.S. markets the company more commonly charges commissions directly to customers. The platform is aimed at more active retail investors, including users seeking options tools, extended-hours trading and real-time market data. For 2025, Webull reported total revenue of US$571.0 million, up from US$390.2 million in 2024. Equity and option order-flow rebates accounted for US$304.1 million, or 53.3% of revenue, making order-flow rebates the company's largest reported revenue category. Interest-related income accounted for US$154.3 million, handling charge income for US$87.3 million and other revenue for US$25.3 million. Options were the larger component of the company's order-flow rebates in 2025, generating US$210.0 million compared with US$94.2 million from equities. Webull also generates revenue from interest-related activities, including margin financing, customer bank deposits, stock lending and corporate bank deposits. The company has stated that its interest-related income is affected by interest rates, customer cash balances, margin balances and demand for stock lending. The company had approximately 20 million registered users worldwide as of February 2024. As of December 31, 2025, it reported 26.8 million registered users, 5.0 million funded accounts and US$24.6 billion in customer assets. As of March 2025, Webull operated in Hong Kong, Singapore, Australia, South Africa, Japan, the United Kingdom, the United States, Indonesia, Canada, Brazil, Thailand, Malaysia and Mexico. == Marketing and sponsorships == Webull has used paid digital advertising, referral incentives, free-stock promotions, affiliate marketing and sports sponsorships to acquire customers and promote its brand. In its 2025 annual filing, the company reported marketing and branding expenses of US$152.3 million in 2023, US$138.7 million in 2024 and US$135.9 million in 2025. Webull said most of its advertising and promotion costs were related to paid search and paid social advertising, and that it had reduced free-stock promotions while shifting toward deposit- and asset-transfer-based incentives. In September 2021, BSE Global, the parent company of the Brooklyn Nets and New York Liberty, entered into a global multi-year agreement with Webull. Under the agreement, Webull became an official sponsor and online brokerage partner of the teams, with branding that included a jersey patch on Brooklyn Nets uniforms. Spo
Internet Security Awareness Training
Internet Security Awareness Training (ISAT) is the training given to members of an organization regarding the protection of various information assets of that organization. ISAT is a subset of general security awareness training (SAT). Even small and medium enterprises are generally recommended to provide such training, but organizations that need to comply with government regulations (e.g., the Gramm–Leach–Bliley Act, the Payment Card Industry Data Security Standard, Health Insurance Portability and Accountability Act, Sarbanes–Oxley Act) normally require formal ISAT for annually for all employees. Often such training is provided in the form of online courses. ISAT, also referred to as Security Education, Training, and Awareness (SETA), organizations train and create awareness of information security management within their environment. It is beneficial to organizations when employees are well trained and feel empowered to take important actions to protect themselves and organizational data. The SETA program target must be based on user roles within organizations and for positions that expose the organizations to increased risk levels, specialized courses must be required. == Coverage == There are general topics to cover for the training, but it is necessary for each organization to have a coverage strategy based on its needs, as this will ensure the training is practical and captures critical topics relevant to the organization. As the threat landscape changes very frequently, organizations should continuously review their training programs to ensure relevance with current trends. Topics covered in ISAT include: Appropriate methods for protecting sensitive information on personal computer systems, including password policy Various computer security concerns, including spam, malware, phishing, social engineering, etc. Consequences of failure to properly protect information, including potential job loss, economic consequences to the firm, damage to individuals whose private records are divulged, and possible civil and criminal law penalties. Being Internet Security Aware means you understand that there are people actively trying to steal data that is stored within your organization's computers. (This often focuses on user names and passwords, so that criminal elements can ultimately get access to bank accounts and other high-value IT assets.) That is why it is important to protect the assets of the organization and stop that from happening. The general scope should include topics such as password security, Email phishing, Social engineering, Mobile device security, Sensitive data security, and Business communications. In contrast, those requiring specialized knowledge are usually required to take technical and in-depth training courses. Suppose an organization determines that it is best to use one of the available training tools on the market, it must ensure it sets objectives that the training can meet, including confirming the training will provide employees with the knowledge to understand risks and the behaviors needed in managing them, actions to take to prevent or detect security incidents, using language easily understandable by the trainees, and ensuring the pricing is reasonable. Organizations are recommended to base ISAT training content on employee roles and their culture; the policy should guide that training for all employees and gave the following as examples of sources of reference materials: National Institute of Standards and Technology (NIST) Special Publication 800-50, Building an Information Technology Security Awareness and Training Program International Standards Organization (ISO) 27002:2013, Information technology—Security techniques—Code of practice for information security controls International Standards Organization (ISO) 27001:2013, Information technology — Security techniques — Information security management systems COBIT 5 Appendix F.2, Detailed Guidance: Services, Infrastructure and Applications Enabler, Security Awareness The training must focus on current threats specific to an organization and the impacts if that materializes as a result of user actions. Including practical examples and ways of dealing with scenarios help users know the appropriate measures to take. It is a good practice to periodically train customers of specific organizations on threats they face from people with malicious intentions. Coverage strategy for SAT should be driven by an organization's policy. It can help truly determine the level of depth of the training and where it should be conducted at a global level or business unit level, or a combination of both. A policy also empowers a responsible party within the organization to run the training. == Importance == Studies show that well-structured security awareness training can significantly reduce the likelihood of cyber incidents caused by human error. According to the Ponemon Institute, organizations that implement regular security training experience up to 70% fewer successful phishing attacks. Additionally, a 2023 Verizon Data Breach Investigations Report found that 74% of breaches involve the human element, highlighting the need for continuous education. Employees are key in whether organizations are breached or not; there must be a policy on creating awareness and training them on emerging threats and actions to take in safeguarding sensitive information and reporting any observed unusual activity within the corporate environment. Research has shown that SAT has helped reduce cyber-attacks within organizations, especially when it comes to phishing, as trainees learned to identify these attack modes and give them the self-assurance to take action appropriately. There is an increase in phishing attacks, and it has become increasingly important for people to understand how to these attacks work, and the actions required to prevent these and SAT has shown a significant impact on the number of successful phishing attacks against organizations. == Compliance Requirements == Various regulations and laws mandate SAT for organizations in specific industries, including the Gramm–Leach–Bliley Act (GLBA) for the financial services, the Federal Information Security Modernization Act of 2014 for federal agencies, and the European Union's General Data Protection Regulation (GDPR). === Federal Information Security Modernization Act === Employees and contractors in federal agencies are required to receive Security Awareness Training annually, and the program needs to address job-related information security risks linked that provide them with the knowledge to lessen security risks. === Health Insurance Portability and Accountability Act === The Health Insurance Portability and Accountability Act has the Security Rule, and Privacy Rule requiring the creation of a security awareness training program and ensuring employees are trained accordingly. === Payment Card Industry Data Security Standard === The Payment Card Industry Security Standards Council, the governing council for stakeholders in the payment industry, formed by American Express, Discover, JCB International, MasterCard, and Visa that developed the DSS as a requirement for the payment industry. Requirement 12.6 requires member organizations to institute a formal security awareness program. There is a published guide for organizations to adhere to when setting up the program. === US States Training Regulations === Some States mandate Security Awareness Training whiles other do not but simply recommend voluntary training. Among states that require the training for its employees include: Colorado (The Colorado Information Security Act, Colorado Revised Statutes 24-37.5-401 et seq.) Connecticut (13 FAM 301.1-1 Cyber Security Awareness Training (PS800)) Florida (Florida Statutes Chapter 282) Georgia (Executive Order GA E.O.182 mandated training within 90 days of issue) Illinois (Cook County) Indiana (IN H 1240) Louisiana (Louisiana Division of Administration, Office of Technology Services p. 52: LA H 633) Maryland (20-07 IT Security Policy) Montana (Mandatory cyber training for executive branch state employees) Nebraska Nevada (agency-by-agency state employee requirement - State Security Standard 123 – IT Security) New Hampshire New Jersey ( NJ A 1654) North Carolina Ohio (IT-15 - Security Awareness and Training) Pennsylvania Texas Utah Vermont Virginia West Virginia (WV Code Section 5A-6-4a) == Training Techniques == Below are some common training techniques, even though some can be blended depending on the operating environment: Interactive video training – This technique allows users to be trained using two-way interactive audio and video instruction. Web-based training – This method allows employees or users to take the training independently and usually has a testing component to determine if learning has taken place. If not, users can be allowed to retake the course and test to ensure there is a complete understanding
BiP (software)
BiP is a freeware instant messaging application developed by Lifecell Ventures Cooperatief U.A., a subsidiary of Turkcell incorporated in the Netherlands. It allows users to send text messages, voice messages and video calling, and it can be downloaded from the App Store, Google Play, and Huawei AppGallery. BiP has over 53 million users worldwide, and was first released in 2013. == Functions == BiP is a secure, and free communication platform. BiP allows making video and audio calls, allows sharing images, videos and location. BiP includes instant translations to 106 languages and exchange rates. President Erdoğan's Communications Office opposed WhatsApp's enforcement of its updated privacy policy and announced that Erdoğan left WhatsApp and opened an account in Telegram and BiP. The Turkish Ministry of National Defense has announced that it will move information groups to BiP for the same reason. == Others == Banglalink announced a BiP messenger partnership in Bangladesh The Communications Office of President Erdoğan opposed WhatsApp's enforcement of its updated privacy policy and announced that Erdoğan left WhatsApp and opened an account in Telegram and BiP. The Turkish Ministry of National Defense has announced that it will move information groups to BiP for the same reason. The CEO of BiP is Burak Akinci. The number of downloads of the app is 80 million globally.
Index locking
In databases an index is a data structure, part of the database, used by a database system to efficiently navigate access to user data. Index data are system data distinct from user data, and consist primarily of pointers. Changes in a database (by insert, delete, or modify operations), may require indexes to be updated to maintain accurate user data accesses. Index locking is a technique used to maintain index integrity. A portion of an index is locked during a database transaction when this portion is being accessed by the transaction as a result of attempt to access related user data. Additionally, special database system transactions (not user-invoked transactions) may be invoked to maintain and modify an index, as part of a system's self-maintenance activities. When a portion of an index is locked by a transaction, other transactions may be blocked from accessing this index portion (blocked from modifying, and even from reading it, depending on lock type and needed operation). Index Locking Protocol guarantees that phantom read phenomenon won't occur. Index locking protocol states: Every relation must have at least one index. A transaction can access tuples only after finding them through one or more indices on the relation A transaction Ti that performs a lookup must lock all the index leaf nodes that it accesses, in S-mode, even if the leaf node does not contain any tuple satisfying the index lookup (e.g. for a range query, no tuple in a leaf is in the range) A transaction Ti that inserts, updates or deletes a tuple ti in a relation r must update all indices to r and it must obtain exclusive locks on all index leaf nodes affected by the insert/update/delete The rules of the two-phase locking protocol must be observed. Specialized concurrency control techniques exist for accessing indexes. These techniques depend on the index type, and take advantage of its structure. They are typically much more effective than applying to indexes common concurrency control methods applied to user data. Notable and widely researched are specialized techniques for B-trees (B-Tree concurrency control) which are regularly used as database indexes. Index locks are used to coordinate threads accessing indexes concurrently, and typically shorter-lived than the common transaction locks on user data. In professional literature, they are often called latches.