This work is devoted to the theoretical and numerical derivation of the moving average coefficients for a first-order autoregressive random field $\{X(\mathbf{t}),\, \mathbf{t}\in \mathbb{Z}^{d}\}$ where $d\geq 2$ and $\mathbb{Z}^d$ is the lattice of points with integer coordinates in the d-dimensional Euclidean space. We develop formulations for the autocorrelation function, the forecast recipe and the forecast error. A sufficient condition for causality is also provided, in addition to an algorithm and corresponding Wolfram Mathematica code for the numerical computation of the moving average coefficients.

Recent investigations have argued that there is a simple explicit representation for the Kolmogorov constant c associated with the subcritical Galton–Watson branching process. We exhibit examples showing that although this representation can be valid, it more often is not. Our work is presented in terms of the limiting conditional mean population size $\mu=c^{-1}$. The analogous quantity for the Markov branching process is denoted by $\widehat\mu$. We show that the simple representation put forward for $\widehat\mu$ in fact is an upper bound that is attained only if the offspring-number probability-generating function is quadratic. The conditional mean $\mu$ is the limit of a computable increasing sequence $(\mu_n$). Estimates of n are determined ensuring that, for any small positive number $\varepsilon$, $0\lt\mu-\mu_n\le \varepsilon$.

In the classical model of diffusion limited aggregation (DLA), introduced by Witten and Sander, the process begins with a single-particle cluster placed at the origin of a space. Then, one at a time, particles make a random walk from infinity until they halt by colliding with the existing cluster. We consider an analogous version of this process on large but finite graphs with a designated source and sink vertex. Initially the cluster of halted particles contains a single particle at the sink vertex. Starting one at a time from the source, each particle makes a random walk in the direction of the sink vertex. The particle halts at the last unoccupied vertex before the walk enters the cluster for the first time, thus increasing the size of the cluster. This continues until the source vertex becomes occupied, at which point the process ends. We study this DLA process on several classes of layered graphs, including Cayley trees of branching factor at least two with a sink vertex attached to the leaves. We determine the finish time of the process for the given classes of graphs and show that the subcomponent of the final cluster linking source to sink is essentially a unique path.

Web-enabled large language models (LLMs) frequently answer queries without crediting the web pages they consume, creating an “attribution gap” in responsible artificial intelligence (AI) usage—defined as the difference between relevant URLs read and those actually cited. Drawing on approximately 14,000 real-world LMArena conversation logs with search-enabled LLM systems, we document three exploitation patterns: (1) no search: 34% of Google Gemini and 24% of OpenAI GPT-4o responses are generated without explicitly fetching any online content; (2) no citation: Gemini provides no clickable citation source in 92% of answers; (3) high-volume, low-credit: Perplexity’s Sonar visits approximately 10 relevant pages per query but cites only three to four. A negative binomial hurdle model shows that the average query answered by Gemini or Sonar leaves about three relevant websites uncited, whereas GPT-4o’s tiny uncited gap is best explained by its selective log disclosures rather than by better attribution. Citation efficiency—extra citations provided per additional relevant web page visited—varies widely across models, from 0.19 to 0.45 on identical queries, underscoring that retrieval design, not technical limits, shapes ecosystem impact. To advance auditing and monitoring of AI systems, we recommend a transparent LLM search architecture based on standardized telemetry and full disclosure of search traces and citation logs.

A fieldworker got more involved in research than intended when he contracted a Giardia duodenalis infection shortly after collecting faecal samples from wild Norwegian reindeer. Almost 50% of the reindeer samples showed heavy infections with G. duodenalis assemblage AI. Molecular comparison with the fieldworker’s infection revealed identical sequences at the loci successfully amplified. Although causality is inherently difficult to establish in wildlife-associated infections, the worker’s long history without previous infection, his intense exposure during sampling, absence of alternative known risk factors, and onset of symptoms consistent with exposure indicate that the reindeer samples were the most plausible source. These findings suggest a rare case of well-supported wildlife-associated Giardia transmission.

Ferry quays experience rapid deterioration due to their exposure to harsh maritime environments and operational ferry impacts. Vibration-based structural health monitoring offers a valuable approach to assessing structural integrity and enhancing the understanding of the structural implications of these impacts. However, practical limitations often restrict sensor placement at critical locations. Consequently, virtual sensing techniques become essential for establishing a Digital Twin and estimating the structural response. This study investigates the application of the Gaussian Process Latent Force Model (GPLFM) for virtual sensing through Bayesian inference on the Magerholm ferry quay, combining in-operation experimental data collected during a ferry impact with a detailed physics-based model. The proposed Physics-Encoded Machine Learning model integrates a reduced-order structural model with a data-driven GPLFM representing the unknown impact forces via their modal contributions. This formulation enables joint probabilistic inference of system states and unmeasured responses while accounting for modeling and measurement uncertainties. Results show that the GPLFM provides accurate posterior mean acceleration response estimates at most locations, even under simplifying modeling assumptions such as linear time-invariant behavior during the impact phase. Lower accuracy was observed at locations in the impact zone. A numerical study was conducted to explore an optimal real-world sensor placement strategy using a Backward Sequential Sensor Placement approach. Sensitivity analyses examined the influence of measurement noise, sensor types, incorrectly assumed damping ratios, and sampling frequencies. The results suggest that the GP latent forces can help accommodate modeling and measurement uncertainties, maintaining acceptable estimation accuracy across scenarios.

Consider n points independently sampled from a density p of class $\mathcal{C}^2$ on a smooth compact d-dimensional submanifold $\mathcal{M}$ of $\mathbb{R}^m$, and consider the random walk visiting these points according to a transition kernel K. We study the almost sure uniform convergence of the generator of this process to the diffusive Laplace–Beltrami operator when n tends to infinity, from which we establish the convergence of the random walk to a diffusion process on the manifold. In contrast to known results, our result does not require the kernel K to be continuous, which covers the cases of walks exploring k-nearest neighbor (kNN) and geometric graphs, and convergence rates are given. The distance between the random walk generator and the limiting operator is separated into several terms: a statistical term, related to the law of large numbers, is treated with concentration tools and an approximation term that we control with tools from differential geometry. The case of kNN Laplacians is detailed. The convergence of the stochastic processes having these operators as generators is also studied, by establishing additional tightness results of their distributions on the space of càdlàg functions.

This paper extends the traditional group self-annuitisation framework by explicitly incorporating mortality heterogeneity among participants. Heterogeneity stems from multiple factors that lead individuals to age at different paces, despite being born in the same year. Ageing is modelled as a finite-state continuous-time Markov process where each state represents a distinct phase of physiological deterioration, and transitions capture the stochastic progression towards death. Benefits are differentiated by ageing state and, after issue, they are dynamically adjusted in response to the realised evolution of both ageing and mortality. Our design is novel in its use of the Markov ageing framework within a risk-sharing scheme and in how benefits are updated. Indeed, both benefits and their respective adjustment coefficients are state-specific. Through the explicit modelling of cross-subsidies across states, the design ensures that actuarial equivalence between benefits and available resources is preserved both at the pool level and within each ageing state. However, we find that benefit adjustments based on actuarial equivalence may display undesirable patterns in some ageing classes, when their size shrinks substantially; this happens, in particular, in the younger ageing states, which are likely to empty out. To contrast such effects, we introduce a design preserving a target level of differentiation across states that mitigates the unfavourable impact of a declining size for younger ages. In our analysis, we point out that such a design (which is desirable in many respects) implies solidarity effects across states. Such effects can be identified by comparing benefit amounts under the two assumptions (i.e., benefits adjusted according to actuarial equivalence or so to preserve a predefined level of differentiation). The proposed framework is tested using Australian mortality data.

The distribution of a random initial age (age composition) of an item is crucial for obtaining its remaining lifetime. A random initial age naturally arises when, for example, an item is drawn at random from a population of continuously manufactured and incepted into operation items. We consider heterogeneous populations of items with lifetime distributions indexed by a frailty parameter. We study different stochastic comparisons for the random age and the remaining (residual) lifetime for items from these populations. The ageing properties for the age composition and remaining lifetime are also discussed. Some examples are provided.

Hamiltonian Monte Carlo (HMC) is a very popular collection of Markov chain Monte Carlo (MCMC) algorithms. One explanation for the popularity of HMC algorithms is their excellent performance as the dimension d of the target becomes large: theoretical analyses show that popular versions of HMC can have a running time that scales as well as $d^{0.25}$ in good conditions, while even an optimally tuned random-walk metropolis (RWM) algorithm will not do better than d. In this paper, we investigate a different scaling question: does HMC beat RWM for targets with well-separated modes? We find that the answer is often no. Our main tool for answering this question is a novel and simple formula for the conductance of HMC based on Liouville’s theorem, and we also show how this new formula can be used to give very short proofs of results that seem tedious to show with the usual formula. We also use this result to compute the spectral gap of HMC algorithms, for both the classical HMC with isotropic momentum and the recent Riemannian HMC, for multimodal targets. While we focus on the concrete comparison of RWM and HMC, we expect qualitatively similar conclusions to hold for other gradient-based algorithms.

We investigate why conservative online news media are often seen as niche, whereas liberal outlets have ideologically broader audiences. We examine two explanatory mechanisms for this asymmetry. The behavioral explanation focuses on differences in homophily, where one ideological camp would be exposed to more cross-cutting content due to more diverse networking preferences. The structural explanation highlights how a platform’s user base places some in the minority, naturally exposing them to more cross-cutting content. We analyze network exposure and sharing of news media content among 420,000 US Twitter users in 2022, prior to Musk’s acquisition of the platform. We find that conservative users, as the minority, were overexposed to cross-cutting media content through their network contacts, while liberal users, as the majority, were underexposed. Consequently, liberal media were shared across party lines, while conservative media were overlooked by liberals and circulated mostly within a tight network of conservative accounts. This apparent paradox suggests that although conservatives primarily engage with their own media, liberal outlets attract a broader audience, including many conservatives. By combining observational data with simulated benchmarks, we find that the structural mechanism plays a primary role in the observed asymmetry, as exposure to liberal content extends farther into conservative online communities.

This work studies time averages of an observable $h(t,X_t)$, where $X_t$ is the solution to a time-inhomogeneous stochastic differential equation (SDE) driven by drift, b(t, x), and diffusion, $\sigma(t{,}{\kern.5pt}x)$, that change sufficiently slowly in time. In this quasistatic regime we derive an approximation to the time average that is computable from properties of the time-homogeneous SDEs driven by $b(t,\cdot)$ and $\sigma(t,\cdot)$ with fixed t; specifically, we utilize $\log$-Sobolev inequalities for the instantaneous invariant distribution and generator for each t. We obtain explicit non-asymptotic error bounds on this quasistatic approximation, both in the form of concentration inequalities and bounds on the expected value. The error bounds demonstrate a competition between the speed of convergence to the instantaneous invariant distributions and their rate of change, matching the intuition that underlies the quasistatic approximation.

We propose a high-dimensional extension of the heteroscedasticity test proposed in Newey and Powell (1987). Our test is based on expectile regression in the proportional asymptotic regime where $n/p \to \delta \in (0,1]$. The asymptotic analysis of the test statistic uses the approximate message passing algorithm, from which we obtain the limiting distribution of the test and establish its asymptotic power. The numerical performance of the test is validated through an extensive simulation study. As real-data applications, we present the analysis based on “international economic growth” data (Belloni et al., 2013), which is found to be homoscedastic, and “supermarket” data (Lan et al., 2016), which is found to be heteroscedastic.

This paper focuses mainly on the Euler scheme of stochastic delay differential equations with locally Lipschitz coefficients. The convergence in probability of the Euler scheme and the corresponding weak limit process of the normalized error process are derived. Furthermore, this paper also considers a class of specific degenerate stochastic delay equations and obtains the associated weak limit process for the stronger error process. The error parameter of this stronger error process for such a degenerate system is n instead of $\sqrt{n}$ in the normalized error process. This causes substantial challenges in the analysis and proofs and the weak limit process also becomes more complicated and involves some additional terms. This result is new and interesting even for the non-delay case.

Greenwashing poses a significant challenge to the fight against climate change by undermining trust in corporate sustainability claims. This study introduced the greenwashing tendency score (GTS), an automatable method designed to detect greenwashing tendencies in corporate sustainability reports. By leveraging textual sentiment and alignment analysis techniques in conjunction with environmental, social, and governance ratings, the GTS quantifies discrepancies between communicated and actual sustainability performance. We applied our methodology to 36 German stock index companies during the years from 2020 to 2022. Our key findings reveal substantial variations in greenwashing tendencies among these companies, emphasizing the need for more transparent and reliable sustainability reporting. The GTS emerged as a scalable, reproducible, and objective tool that can aid, for example, investors, regulators, and Non-government organizations in identifying greenwashing practices. This research contributed to the sustainable finance literature by introducing a neutral and open measure to assess firms’ greenwashing tendency, summarizing implications for policymaking and regulatory authorities and discussing its potential for long-term accountability and integrity in corporate sustainability communications.

Enterovirus A71 was first isolated in California in 1969, with the earliest retrospective detection traced back to 1963 in the Netherlands, but its early spread remains unclear. Using age-specific seroprevalence data from children aged 1–10 years in Kawasaki City, Japan, collected annually from 1966–1973, we applied serocatalytic models to estimate annual force of infection during 1959–1973. Several models were tested, incorporating different assumptions about time-varying force of infection, age-dependent susceptibility, and seroreversion, to identify the best fit to the data. Model comparison identified the models with independent annual infection probability or two distinct outbreak periods, both including age-dependent force of infection and seroreversion, as optimal. All top models consistently identified two major transmission periods: 1961–1962 and 1968–1969. The two-outbreak model estimated mean attack rates of 21.8% and 37.8% for the earlier and later outbreaks under seroreversion, and 19.8% and 34.9% under age-dependent force of infection. Our findings provide evidence of enterovirus A71 circulation in Japan during two distinct periods in the 1960s, coinciding with early detections in Europe and the USA, suggesting global distribution by that decade. This study underscores the value of testing archived sera for reconstructing pathogen emergence and spread.

In classical credibility theory, estimation is typically limited to the hypothetical mean, restricting its use for premium principles that depend on higher-order moments. To address this, we develop a credibility-based framework for estimating the process variance under both known and unknown hypothetical means and apply these estimators to a broad class of variance-related premium principles, including the expected value, variance, standard deviation, and modified-variance principles. The estimators are derived via constrained linear projection techniques, minimizing the mean squared error between the estimator and the true process variance. Explicit formulas are obtained that are optimal among affine transformations of the data. The proposed estimators exhibit desirable statistical properties, including conditional unbiasedness, consistency, mean squared error convergence, and asymptotic normality. Numerical studies demonstrate their favorable convergence behavior, and an empirical analysis with real insurance data highlights their practical relevance. This framework extends Bühlmann’s classical credibility theory to second-moment estimation while remaining computationally tractable and requiring only mild moment conditions, without specifying the population or prior distributions.