Environmental data science for spatial extremes has traditionally relied heavily on max-stable processes. Even though the popularity of these models has perhaps peaked with statisticians, they are still perceived and considered as the “state of the art” in many applied fields. However, while the asymptotic theory supporting the use of max-stable processes is mathematically rigorous and comprehensive, we think that it has also been overused, if not misused, in environmental applications, to the detriment of more purposeful and meticulously validated models. In this article, we review the main limitations of max-stable process models, and strongly argue against their systematic use in environmental studies. Alternative solutions based on more flexible frameworks using the exceedances of variables above appropriately chosen high thresholds are discussed, and an outlook on future research is given. We consider the opportunities offered by hybridizing machine learning with extreme-value statistics, highlighting seven key recommendations moving forward.

Let θ > 0 be a measure of the average step size of a stochastic process {pn(θ) }n=1(∞). Conditions are given under which pn(θ) is approximately normally distributed when n is large and θ is small. This result is applied to a number of learning models where θ is a learning rate parameter and pn(θ) is the probability that the subject makes a certain response on the nth experimental trial. Both linear and stimulus sampling models are considered.

In the continuous response (CR) task, the subject is given the name of a familiar category and must respond with category examples within a fixed time limit. The response sequence and the interresponse times are the measured variables. A general model is proposed for performance in the CR task. The time taken to generate a response is identified with the shortest search time from among a set of simultaneous search processes. Each response has a generation probability that is a function of the parameters governing the search process durations. Data involving the category “states in the United States” are used to test the model. The model accounts for the performance of nearly all individuals in these data.

Any family of simple response time distributions that correspond to different values of stimulation variables can be modeled by a deterministic stimulation-dependent process that terminates when it crosses a randomly preset criterion. The criterion distribution function is stimulation-independent and can be chosen arbitrarily, provided it is continuous and strictly increasing. Any family of N-alternative choice response time distributions can be modeled by N such process-criterion pairs, with response choice and response time being determined by the process that reaches its criterion first. The joint distribution of the N criteria can be chosen arbitrarily, provided it satisfies certain unrestrictive conditions. In particular, the criteria can be chosen to be stochastically independent. This modeling scheme, therefore, is a descriptive theoretical language rather than an empirically falsifiable model. The only role of the criteria in this theoretical language is to numerically calibrate the ordinal-scale axes for the deterministic response processes.

Researchers are often interested in testing for measurement invariance with respect to an ordinal auxiliary variable such as age group, income class, or school grade. In a factor-analytic context, these tests are traditionally carried out via a likelihood ratio test statistic comparing a model where parameters differ across groups to a model where parameters are equal across groups. This test neglects the fact that the auxiliary variable is ordinal, and it is also known to be overly sensitive at large sample sizes. In this paper, we propose test statistics that explicitly account for the ordinality of the auxiliary variable, resulting in higher power against “monotonic” violations of measurement invariance and lower power against “non-monotonic” ones. The statistics are derived from a family of tests based on stochastic processes that have recently received attention in the psychometric literature. The statistics are illustrated via an application involving real data, and their performance is studied via simulation.

A functional traits approach was adopted to examine patterns of fish diversity in relation to environmental and spatial variables and for comparison with findings from earlier analyses of fish taxonomic diversity in a nearly pristine river in the Llanos region of Colombia. Fishes were surveyed during the low-water period at 34 sites along the longitudinal fluvial gradient of the Bita River, a clearwater tributary of the Orinoco River. We compiled a matrix of 37 traits associated with locomotion, feeding, and defense for 132 species to compute functional β diversity metrics among four reaches along the entire longitudinal fluvial gradient and also within reaches. Functional redundancy (trait under-dispersion) was found to be common throughout the fluvial gradient. Functional β diversity was high at both spatial scales, with widespread evidence of functional nestedness and functional turnover, especially in upper reaches. In the Bita River, environmental filtering and stochastic processes, such as random dispersal, appear to be the dominant mechanisms influencing fish functional diversity. Inferences derived from analysis of functional versus taxonomic diversity are largely congruent, although analysis of functional β diversity found more evidence of nestedness than turnover at both spatial scales. Both lines of evidence stress the importance of habitat heterogeneity and aquatic habitat connectivity for conservation of high species diversity in this system.

The general theory developed by Ben Yaacov for metric structures provides Fraïssé limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.

We investigate the concept of orbital free entropy from the viewpoint of the matrix liberation process. We will show that many basic questions around the definition of orbital free entropy are reduced to the question of full large deviation principle for the matrix liberation process. We will also obtain a large deviation upper bound for a certain family of random matrices that is essential to define the orbital free entropy. The resulting rate function is made up into a new approach to free mutual information.

The Quicksort process R (Rösler (2018)) can be characterized as the unique endogenous solution of the inhomogeneous stochastic fixed point equation R=D(UR1(1∧t∕U)+𝟭{U<t}(1-U)R2((t-U)∕(1-U))+C(U,t))t on the space 𝒟 of càdlàg functions, such that R(1) has the Quicksort distribution. In this paper we characterize all 𝒟-valued solutions of that equation. Every solution can be represented as the convolution of a solution of the inhomogeneous equation and a general solution of the homogeneous equation (Rüschendorf (2006)). The general solutions of the homogeneous equation are the distributions of Cauchy processes Y with constant drift. Any distribution of R+Y for independent R and Y is a solution of the inhomogeneous equation. Every solution of the inhomogeneous equation is of the form R+Y, where R and Y are independent. The endogenous solutions for the inhomogeneous equation are the shifted Quicksort process distributions. In comparison, the Quicksort distribution is the endogenous solution of the Quicksort fixed point equation unique up to a constant (Rösler (1991)). The general solution can be represented as the convolution of the shifted Quicksort distribution and some symmetric Cauchy distribution (Fill and Janson (2000)), possibly degenerate.

We construct a response adaptive design, described in terms of a two-color urn model, targeting fixed asymptotic allocations. We prove asymptotic results for the process of colors generated by the urn and for the process of its compositions. An application of the proposed urn model is presented in an estimation problem context.

One of the central results in Einstein’s theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation.

The ‘coupon collection problem’ refers to a class of occupancy problems in which j identical items are distributed, independently and at random, to n cells, with no restrictions on multiple occupancy. Identifying the cells as coupons, a coupon is ‘collected’ if the cell is occupied by one or more of the distributed items; thus, some coupons may never be collected, whereas others may be collected once or twice or more. We call the number of coupons collected exactly r times coupons of type r. The coupon collection model we consider is general, in that a random number of purchases occurs at each stage of collecting a large number of coupons; the sample sizes at each stage are independent and identically distributed according to a sampling distribution. The joint behavior of the various types is an intricate problem. In fact, there is a variety of joint central limit theorems (and other limit laws) that arise according to the interrelation between the mean, variance, and range of the sampling distribution, and of course the phase (how far we are in the collection processes). According to an appropriate combination of the mean of the sampling distribution and the number of available coupons, the phase is sublinear, linear, or superlinear. In the sublinear phase, the normalization that produces a Gaussian limit law for uncollected coupons can be used to obtain a multivariate central limit law for at most two other types — depending on the rates of growth of the mean and variance of the sampling distribution, we may have a joint central limit theorem between types 0 and 1, or between types 0, 1, and 2. In the linear phase we have a multivariate central limit theorem among the types 0, 1,…, k for any fixed k.

We use the recently proposed Nested Stochastic Simulation Algorithm (Nested SSA) to simulate the cell cycle model for budding yeast. The results show that Nested SSA is able to significantly reduce the computational cost while capturing the essential dynamical features of the system.

One shows that the linearized Navier-Stokes equation in ${\mathcal{O}}{\subset} R^d,\;d \ge 2$ , around an unstable equilibriumsolution is exponentially stabilizable in probability by aninternal noise controller $V(t,\xi)=\displaystyle\sum\limits_{i=1}^{N} V_i(t)\psi_i(\xi)\dot\beta_i(t)$ , $\xi\in{\mathcal{O}}$ , where $\{\beta_i\}^N_{i=1}$ areindependent Brownian motions in a probability space and $\{\psi_i\}^N_{i=1}$ is a system of functions on ${\mathcal{O}}$ withsupport in an arbitrary open subset ${\mathcal{O}}_0\subset {\mathcal{O}}$ . Thestochastic control input $\{V_i\}^N_{i=1}$ is found in feedbackform. One constructs also a tangential boundary noise controllerwhich exponentially stabilizes in probability the equilibriumsolution.

Face à la nécessité d'accroître les performances de mobilité tout terrain de ses véhicules et à sa volonté d'y intégrer désormais des équipements "low cost" provenant du secteur civil, la Société Nexter s'est dotée de démarches de conception sécurisantes, permettant de faire face à ce type de compromis antinomique.Basée sur la capacité d'évaluer rapidement les valeurs des contraintes extrêmes produites sur les structures embarquées, dès le début du cycle de développement du produit, l'originalité de la démarche de conception proposée réside dans le fait qu'elle permet de synthétiser un grand nombre de processus d'excitation stochastique, décrits par le profil d'emploi opérationnel des clients, prendre en compte le caractère aléatoire des excitations concernées, en supposant de surcroît une indépendance statistique des situations stochastiques retenues, et de revenir à un chargement dynamique unique, reproduisant fidèlement les valeurs des contraintes extrêmes des processus concernés, avec un risque de dépassement maîtrisé puisque spécifié.Éprouvée dans le cadre du développement du VBCI (Véhicule Blindé de Combat d'Infanterie), cette démarche de conception dynamique a notamment mis en évidence le caractère peu sécurisant des démarches de conception forfaitaire à caractère statique déployées dans le passé.

The ‘square root formula’ in the Internet transmission control protocol (TCP) states that if the probability p of packet loss becomes small and there is independence between packets, then the stationary distribution of the congestion window W is such that the distribution of W√p is almost independent of p and is completely characterizable. This paper gives an elementary proof of the convergence of the stationary distributions for a much wider class of processes that includes classical TCP as well as T. Kelly's ‘scalable TCP’. This paper also gives stochastic dominance results that translate to a rate of convergence.