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In this article, we derive (local) orthogonality graphs for the popular continuous‐time state space models, including in particular multivariate continuous‐time ARMA (MCARMA) processes. In these (local) orthogonality graphs, vertices represent the components of the process, directed edges between the vertices indicate causal influences and undirected edges indicate contemporaneous correlations between the component processes. We present sufficient criteria for state space models to satisfy the assumptions of Fasen‐Hartmann and Schenk (2024a) so that the (local) orthogonality graphs are well‐defined and various Markov properties hold. Both directed and undirected edges in these graphs are characterised by orthogonal projections on well‐defined linear spaces. To compute these orthogonal projections, we use the unique controller canonical form of a state space model, which exists under mild assumptions, to recover the input process from the output process. We are then able to derive some alternative representations of the output process and its highest derivative. Finally, we apply these representations to calculate the necessary orthogonal projections, which culminate in the characterisations of the edges in the (local) orthogonality graph. These characterisations are given by the parameters of the controller canonical form and the covariance matrix of the driving Lévy process. [ABSTRACT FROM AUTHOR]

Why do we remember the past, and plan the future? We introduce a toy model in which to investigate emergent time asymmetries: the causal multibaker maps. These are reversible discrete-time dynamical systems with configurable causal interactions. Imposing a suitable initial condition or "Past Hypothesis", and then coarse-graining, yields a Pearlean locally causal structure. While it is more common to speculate that the other arrows of time arise from the thermodynamic arrow, our model instead takes the causal arrow as fundamental. From it, we obtain the thermodynamic and epistemic arrows of time. The epistemic arrow concerns records, which we define to be systems that encode the state of another system at another time, regardless of the latter system's dynamics. Such records exist of the past, but not of the future. We close with informal discussions of the evolutionary and agential arrows of time, and their relevance to decision theory. [ABSTRACT FROM AUTHOR]

Discovering causal relationships from observational data is a challenging task that relies on assumptions connecting statistical quantities to graphical or algebraic causal models. In this work, we focus on widely employed assumptions for causal discovery when objects of interest are (multivariate) groups of random variables rather than individual (univariate) random variables, as is the case in a variety of problems in scientific domains such as climate science or neuroscience. If the group level causal models are derived from partitioning a micro-level model into groups, we explore the relationship between micro- and group level causal discovery assumptions. We investigate the conditions under which assumptions like causal faithfulness hold or fail to hold. Our analysis encompasses graphical causal models that contain cycles and bidirected edges. We also discuss grouped time series causal graphs and variants thereof as special cases of our general theoretical framework. Thereby, we aim to provide researchers with a solid theoretical foundation for the development and application of causal discovery methods for variable groups.

This paper is concerned with approximations of invariant probability measures for stochastic functional differential equations (SFDEs) using a backward Euler-Maruyama (BEM) scheme under one-sided Lipschitz condition on the drift coefficient. Firstly, the strong convergence of the numerical "segment sequence" from the BEM scheme on finite time interval [ 0 , T ] is established. In addition, it is also demonstrated that the numerical segment sequence from the BEM scheme is a Markov process, and the corresponding discrete-time semigroup generated by this BEM scheme admits a unique numerical invariant probability measure. Finally, it is revealed that the numerical invariant probability measure converges to the underlying one in a Wasserstein distance. [ABSTRACT FROM AUTHOR]

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Return and risk are significant parameters in selecting an optimal portfolio, depending on the portfolio return distribution. In a stochastic process, the Markov property causes the future distribution of a random process to be measurable according to the state-transition matrix and the initial process state. According to the main idea of the present study in the optimal portfolio selection, portfolio weights are chosen in a way that the Markov property is established for the portfolio return series and the distribution of future portfolio returns is close to the distribution of investor’s expected returns; hence, K-L divergence (Kullback–Leibler divergence) is utilized as a criterion of closeness. Using this idea, an optimal portfolio selection model was designed and implemented in the present study. This optimal portfolio was optimized using a Markov approach and according to historical data of 10 indices on the Tehran Stock Exchange from 2009 to 2022 in a six-member state. The optimal portfolio performance evaluation using the Sharpe ratio and value at risk criteria indicated that the research model had a higher performance than the mean-variance and weight parity models. [ABSTRACT FROM AUTHOR]

Why do we remember the past, and plan the future? We introduce a toy model in which to investigate emergent time asymmetries: the causal multibaker maps. These are reversible discrete-time dynamical systems with configurable causal interactions. Imposing a suitable initial condition or “Past Hypothesis”, and then coarse-graining, yields a Pearlean locally causal structure. While it is more common to speculate that the other arrows of time arise from the thermodynamic arrow, our model instead takes the causal arrow as fundamental. From it, we obtain the thermodynamic and epistemic arrows of time. The epistemic arrow concerns records, which we define to be systems that encode the state of another system at another time, regardless of the latter system’s dynamics. Such records exist of the past, but not of the future. We close with informal discussions of the evolutionary and agential arrows of time, and their relevance to decision theory.

Hamza and Klebaner (2007) [10] posed the problem of constructing martingales with one-dimensional Brownian marginals that differ from Brownian motion, so-called fake Brownian motions. Besides its theoretical appeal, this problem represents the quintessential version of the ubiquitous fitting problem in mathematical finance where the task is to construct martingales that satisfy marginal constraints imposed by market data. Non-continuous solutions to this challenge were given by Madan and Yor (2002) [22], Hamza and Klebaner (2007) [10], Hobson (2016) [11] and Fan et al. (2015) [8], whereas continuous (but non-Markovian) fake Brownian motions were constructed by Oleszkiewicz (2008) [23], Albin (2008) [1], Baker et al. (2006) [4], Hobson (2013) [14], Jourdain and Zhou (2020) [16]. In contrast, it is known from Gyöngy (1986) [9], Dupire (1994) [7] and ultimately Lowther (2008) [17] and Lowther (2009) [20] that Brownian motion is the unique continuous strong Markov martingale with one-dimensional Brownian marginals. We took this as a challenge to construct examples of a "barely fake" Brownian motion, that is, continuous Markov martingales with one-dimensional Brownian marginals that miss out only on the strong Markov property. [ABSTRACT FROM AUTHOR]

We study reflecting Brownian motion with drift constrained to a wedge in the plane. Our first set of results provides necessary and sufficient conditions for existence and uniqueness of a solution to the corresponding submartingale problem with drift, and show that its solution possesses the Markov and Feller properties. Next, we study a version of the problem with absorption at the vertex of the wedge. In this case, we provide a condition for existence and uniqueness of a solution to the problem and some results on the probability of the vertex being reached. [ABSTRACT FROM AUTHOR]

The aim of the paper is to carry out survival analysis of a novel multi-state model on infectious disease considering various levels of severity using semi-Markov processes. Various levels of severity of the disease over time and transitions between these severity levels have been considered. Transition probabilities and expected waiting times are derived. Expressions for mean survival time, expected total time in home isolation, and expected total time in hospital are obtained. The analysis of the proposed model is carried out through numerical computation and plotting several graphs. Important conclusions are drawn. The modelling framework proposed here can be used to model any infectious disease irrespective of disease states. The study will be helpful in designing effective measures to control the infectious disease and selecting the appropriate intervention policies. [ABSTRACT FROM AUTHOR]

This paper is concerned with exponential synchronization for a class of coupled neural networks with hybrid delays and stochastic distributed delayed impulses. First of all, based on the average impulsive interval method, total probability formula and ergodic theory, several novel impulsive Halanay differential inequalities are established. Two types of stochastic impulses, i.e., stochastic distributed delayed impulses with dependent property and Markov property have been taken into account, respectively. Secondly, some criteria on exponential synchronization in the mean square of a class of coupled neural networks with stochastic distributed delayed impulses are acquired by combining the proposed lemmas and graph theory. The validity of the theoretical results is demonstrated by several numerical simulation examples.

Crustaceans exhibit discontinuous growth as they shed hard shells periodically. Fundamentally, the growth of crustaceans is typically assessed through two key components, length increase after molting (LI) and time intervals between consecutive molts (TI). In this article, we propose a unified likelihood approach that combines a generalized additive model and a Cox proportional hazard model to estimate the parameters of LI and TI separately in crustaceans. This approach captures the observed discontinuity in individuals, providing a comprehensive understanding of crustacean growth patterns. Our study focuses on 75 ornate rock lobsters (Panulirus ornatus) off the Torres Strait in northeastern Australia. Through a simulation study, we demonstrate the effectiveness of the proposed models in characterizing the discontinuity with a continuous growth curve at the population level.

This paper explores the theory and practice characteristics of management accounting, expands upon the function of management accounting, and outlines the business process of strategic management accounting. Then, it constructs the functional effect analysis model of management accounting, establishes the causality analysis model of the structural equation, explores the markov property of the equation, and uses multiple linear regression equations to estimate the regression parameters. Finally, it analyzes the correlation between the overall level of strategic management accounting, the application level of management accounting, and the horizontal integration level of management accounting and enterprise performance. According to regression analysis, the three dimensions have a significant positive correlation with enterprise performance at the 1% level. The structural equation analysis indicates that the path coefficient of management accounting on performance is 0.388, which is significant at a 1% level. The application of management accounting can have a positive impact on the performance level of enterprises, as shown. This research promotes the innovative practice of management accounting and is of great significance for the high-quality development of enterprises.

The Markov property is widely imposed in analysis of time series data. Correspondingly, testing the Markov property, and relatedly, inferring the order of a Markov model, are of paramount importance. In this article, we propose a nonparametric test for the Markov property in high-dimensional time series via deep conditional generative learning. We also apply the test sequentially to determine the order of the Markov model. We show that the test controls the type-I error asymptotically, and has the power approaching one. Our proposal makes novel contributions in several ways. We utilise and extend state-of-the-art deep generative learning to estimate the conditional density functions, and establish a sharp upper bound on the approximation error of the estimators. We derive a doubly robust test statistic, which employs a nonparametric estimation but achieves a parametric convergence rate. We further adopt sample splitting and cross-fitting to minimise the conditions required to ensure the consistency of the test. We demonstrate the efficacy of the test through both simulations and the three data applications. [ABSTRACT FROM AUTHOR]

Consider the Ising model on a centered box of side length n in Z d with ∓ -boundary conditions that are minus in the upper half-space and plus in the lower half-space. Dobrushin famously showed that in dimensions d ≥ 3 , at low-temperatures the Ising interface (dual-surface separating the plus/minus phases) is rigid, i.e., it has O(1) height fluctuations. Recently, the authors decomposed these oscillations into pillars and identified their typical shape, leading to a law of large numbers and tightness of their maximum. Suppose we condition on a height-h level curve of the interface, bounding a set S ⊂ Z d - 1 , along with the entire interface outside the cylinder S × Z : what does the interface in S × Z look like? Many models of random surfaces (e.g., SOS and DGFF) fundamentally satisfy the domain Markov property, whereby their heights on S only depend on the heights on S c through the heights on ∂ S . The Ising interface importantly does not satisfy this property; the law of the interface depends on the full spin configuration outside S × Z . Here we establish an approximate domain Markov property inside the level curves of the Ising interface. We first extend Dobrushin’s result to this setting, showing the interface in S × Z is rigid about height h, with exponential tails on its height oscillations. Then we show that the typical tall pillars in S × Z are uniformly absolutely continuous with respect to tall pillars of the unconditional Ising interface. Using this we identify the law of large numbers, tightness, and sharp Gumbel tail bounds on the maximum oscillations in S × Z about height h, showing that these only depend on the conditioning through the cardinality of S. [ABSTRACT FROM AUTHOR]

We formulate a probabilistic Markov property in discrete time under a dynamic risk framework with minimal assumptions. This is useful for recursive solutions to risk-sensitive versions of dynamic optimisation problems such as optimal prediction, where at each stage the recursion depends on the whole future. The property holds for standard measures of risk used in practice, and is formulated in several equivalent versions including a representation via acceptance sets, a strong version, and a dual representation. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the number of customers in infinite-server systems driven by Hawkes processes. In these systems, the self-exciting arrival process is assumed to be represented by a Hawkes process and the self-exciting service process by a state-dependent Hawkes process (sdHawkes process). Under some suitable conditions, for the Hawkes / sdHawkes / ∞ system, the Markov property of the system is derived. The joint time-dependent distribution of the number of customers in the system, the arrival intensity and the server intensity is characterized by a system of differential equations. Then, the time-dependent results are also deduced for the M / sdHawkes / ∞ system. [ABSTRACT FROM AUTHOR]

We use data from two representative US household surveys, the Medical Expenditure Panel Survey (MEPS) and the Health and Retirement Study (RAND-HRS) to estimate transition probability matrices between health states over the lifecycle from age 20–95. We compare nonparametric counting methods and parametric methods where we control for individual characteristics as well as time and cohort effects. We align two year transition probabilities from HRS with one-year transition probabilities in MEPS using a stochastic root method assuming a Markov structure. We find that the nonparametric counting method and the regression specifications based on ordered logit models produce similar results over the lifecycle. However, the counting method overestimates the probabilities of transitioning into bad health states. In addition, we find that young women have worse health prospects than their male counterparts but once individuals get older, being female is associated with transitioning into better health states with higher probabilities than men. We do not find significant differences of the conditional health transition probabilities between African Americans and the rest of the population. We also find that the lifecycle patterns are stable over time. Finally, we discuss issues with controlling for time effects, sample attrition, the Markov assumption, and other modeling issues that can arise with categorical outcome variables. [ABSTRACT FROM AUTHOR]

The analysis of air pollution behavior is becoming crucial, where information on air pollution behavior is vital for managing air quality events. Many studies have described the stochastic behavior of air pollution based on the Markov chain (MC) models. Fitting the optimum order of MC models is essential for describing the stochastic process. However, uncertainty remains concerning the optimum order of such models for representing and characterizing air pollution index (API) data. In this study, the optimum order of the MC models for hourly and daily API sequences from seven stations in the central region of Peninsular Malaysia is identified, based on the Bayesian information criteria (BIC), contributing to exploring an adequate explanation of the probabilistic dependence of air pollution. A summary of the statistics for the API was calculated prior to the analysis. The Markov property and the divergence for the empirically estimated transition matrix of an MC sequence are also investigated. It is found from the analysis that the optimum order varies from one station to another. At most stations, for both observed and simulated API data, the second and third orders of the MC models are found to be optimum for hourly API occurrences, while the first-order MC is found to be most fitting for describing the dynamics of the daily API. Overall, fitting the optimum order of the MC model for the API data sequence captured the delay effect of air pollution. Accordingly, we concluded that the air quality standard lies within controllable limits, except for some infrequent occurrences of API values exceeding the unhealthy level. [ABSTRACT FROM AUTHOR]

Multi-state models for event history analysis most commonly assume the process is Markov. This article considers tests of the Markov assumption that are applicable to general multi-state models. Two approaches using existing methodology are considered; a simple method based on including time of entry into each state as a covariate in Cox models for the transition intensities and a method involving detecting a shared frailty through a stratified Commenges-Andersen test. In addition, using the principle that under a Markov process the future rate of transitions of the process at times $t > s$ should not be influenced by the state occupied at time $s$, a new class of general tests is developed by considering summaries from families of log-rank statistics where patients are grouped by the state occupied at varying initial time $s$. An extended form of the test applicable to models that are Markov conditional on observed covariates is also derived. The null distribution of the proposed test statistics are approximated by using wild bootstrap sampling. The approaches are compared in simulation and applied to a dataset on sleeping behavior. The most powerful test depends on the particular departure from a Markov process, although the Cox-based method maintained good power in a wide range of scenarios. The proposed class of log-rank statistic based tests are most useful in situations where the non-Markov behavior does not persist, or is not uniform in nature across patient time. [ABSTRACT FROM AUTHOR]

Suppose that risk reserves of an insurance company are governed by a Markov-modulated classical risk model with parameters modulated by a finite-state irreducible Markov chain. The main purpose of this paper is to calculate ultimate ruin probability that ruin time, the first time when risk reserve is negative, is finite. We apply Banach contraction principle, q-scale functions and Markov property to prove that ultimate ruin probability is the fixed point of a contraction mapping in terms of q-scale functions and that ultimate ruin probability can be calculated by constructing an iterative algorithm to approximate the fixed point. Unlike Gajek and Rudź (Insurance: Mathematics and Economics, 80 (2018), 45–53), our paper uses q-scale functions to obtain more explicit Lipschitz constant in Banach contraction principle in our case so that proofs of several Lemmas and theorems in their Appendix are unnecessary and some of their assumptions are confirmed in our case. [ABSTRACT FROM AUTHOR]

Rating transition models are widely used for credit risk evaluation. It is not uncommon that a time-homogeneous Markov rating migration model will deteriorate quickly after projecting repeatedly for a few periods. This is because the time-homogeneous Markov condition is generally not satisfied. For a credit portfolio, the rating transition is usually path-dependent. In this paper, we propose a recurrent neural network (RNN) model for modeling path-dependent rating migration. An RNN is a type of artificial neural network where connections between nodes form a directed graph along a temporal sequence. There are neurons for input and output at each time period. The model is informed by the past behaviors for a loan along the path. Information learned from previous periods propagates to future periods. The experiments show that this RNN model is robust. [ABSTRACT FROM AUTHOR]

The first-passage probabilities in stochastic processes are important for the evaluation of structural reliability. In this work, Wiener process's first-passage probability was analysed using different methods. First, the Poisson process method was used to analyse the first-passage probability of the Wiener process, and it was found this method cannot calculate the first-passage probability of the Wiener process. Then, the symmetry and Markov property of the Wiener process were used to derive equations for calculating the first-passage probability of a Wiener process. The results indicate analytical expressions for the first-passage time probability distribution function of the Wiener process can be derived either through the symmetry or Markov property of the process. Neither of the analytical expressions requires the use of assumptions in their derivations. The analytical method based on the symmetry of the Wiener process yields fully exact solution, whereas the analytical method based on the Markov property of the Wiener process produces highly accurate solution. Finally, the Wiener process is used to represent the time-varying amount of structural degradation at any point in time, leading to a non-stationary stochastic model for the degradation process, which will provide a basis for an estimation of the durability and reliability of concrete structures. [ABSTRACT FROM AUTHOR]

This article offers a mathematical model for distributed object­oriented stochastic hybrid systems (DOBSHS). DOBSHS are composite objects communicating with other objects through the exchange of messages through an asynchronous medium such as a network. An important component of the model is the probabilistic nature of the DOBSHS, in which the state of the system is described by stochastic differential equations with instantaneous probabilistic state changes when certain conditions are met. Also probabilistic is the nature of the messaging environment, in which the model of message delivery time is a random variable. Such problems are often encountered in practice in various areas and issues of formal modeling and verification of their properties are very important. The article presents a mathematical model of DOBSHS and proved that it has a Markov property.

The analysis of air pollution behavior is becoming crucial, where information on air pollution behavior is vital for managing air quality events. Many studies have described the stochastic behavior of air pollution based on the Markov chain (MC) models. Fitting the optimum order of MC models is essential for describing the stochastic process. However, uncertainty remains concerning the optimum order of such models for representing and characterizing air pollution index (API) data. In this study, the optimum order of the MC models for hourly and daily API sequences from seven stations in the central region of Peninsular Malaysia is identified, based on the Bayesian information criteria (BIC), contributing to exploring an adequate explanation of the probabilistic dependence of air pollution. A summary of the statistics for the API was calculated prior to the analysis. The Markov property and the divergence for the empirically estimated transition matrix of an MC sequence are also investigated. It is found from the analysis that the optimum order varies from one station to another. At most stations, for both observed and simulated API data, the second and third orders of the MC models are found to be optimum for hourly API occurrences, while the first-order MC is found to be most fitting for describing the dynamics of the daily API. Overall, fitting the optimum order of the MC model for the API data sequence captured the delay effect of air pollution. Accordingly, we concluded that the air quality standard lies within controllable limits, except for some infrequent occurrences of API values exceeding the unhealthy level.

In the analysis of the readiness of means of transport, the Markov and semi-Markov processes are particularly applicable, allowing for the description of the usage process over long periods of time, determination of indicators of the exploitability and readiness of the used set of objects, as well as simulation of long-term forecasts of the usage process results. The studies presented in the literature usually concern the theoretical side of the matter, mainly the construction of formal models of the process of changing the operating states of a vehicle. Less attention is paid to the empirical side, especially with regard to the actual conditions of use. Examples of experimental observations presented in the literature most often concern individual cases. This paper lists selected irregularities and presents an example of a study of a real transport system based on semi-Markov processes.

We establish a principle that we call the Fatou-Shishikura injection for Newton maps of polynomials: there is a dynamically natural injection from the set of non-repelling periodic orbits of any Newton map to the set of its critical orbits. This injection obviously implies the classical Fatou-Shishikura inequality, but it is stronger in the sense that every non-repelling periodic orbit has its own critical orbit. Moreover, for every Newton map we associate a forward invariant graph (a puzzle) which provides a dynamically defined partition of the Riemann sphere into closed topological disks (puzzle pieces). This puzzle construction is for rational Newton maps what Yoccoz puzzles are for polynomials: it provides the foundation for all kinds of rigidity results of Newton maps beyond our Fatou-Shishikura injection. Moreover, it gives necessary structure for a classification of the postcritically finite maps in the spirit of Thurston theory. [ABSTRACT FROM AUTHOR]

In the analysis of the readiness of means of transport, the Markov and semi-Markov processes are particularly applicable, allowing for the description of the usage process over long periods of time, determination of indicators of the exploitability and readiness of the used set of objects, as well as simulation of long-term forecasts of the usage process results. The studies presented in the literature usually concern the theoretical side of the matter, mainly the construction of formal models of the process of changing the operating states of a vehicle. Less attention is paid to the empirical side, especially with regard to the actual conditions of use. Examples of experimental observations presented in the literature most often concern individual cases. This paper lists selected irregularities and presents an example of a study of a real transport system based on semi-Markov processes. [ABSTRACT FROM AUTHOR]

Timely diagnostic feedback is helpful for students and teachers, enabling them to adjust their learning and teaching plans according to a current diagnosis. Motivated by the practical concern that the simultaneity estimation strategy currently adopted by longitudinal learning diagnosis models does not provide timely diagnostic feedback, this study proposes a new Markov estimation strategy, which follows the Markov property. A simulation study was conducted to explore and compare the performance of four estimation strategies: the simultaneity, the Markov, the anchor-item, and the separated estimation strategies. The results show that their performance was highly consistent, and they presented in the following relative order: simultaneity > Markov > anchor-item ≥ separated. Overall, although accuracy in parameter estimation is sacrificed slightly with the proposed strategy, it can provide timely diagnostic feedback to practitioners, which is in line with the concept of "assessment for learning" and the needs of formative assessment. [ABSTRACT FROM AUTHOR]

We propose a directed acyclic hypergraph framework for a probabilistic graphical model that we call Bayesian hypergraphs. The space of directed acyclic hypergraphs is much larger than the space of chain graphs. Hence Bayesian hypergraphs can model much finer factorizations than Bayesian networks or LWF chain graphs and provide simpler and more computationally efficient procedures for factorizations and interventions. Bayesian hypergraphs also allow a modeler to represent causal patterns of interaction such as Noisy-OR graphically (without additional annotations). We introduce global, local and pairwise Markov properties of Bayesian hypergraphs and prove under which conditions they are equivalent. We also extend the causal interpretation of LWF chain graphs to Bayesian hypergraphs and provide corresponding formulas and a graphical criterion for intervention. [ABSTRACT FROM AUTHOR]

Habit formation models are widely used in macroeconomics and finance. However, little work has been done to characterize habit formation. In this paper, we provide a simple axiomatic foundation of linear habit formation models that satisfy the Markov (or memoryless) property. We also provide a method to characterize a general class of time-nonseparable but additive models that satisfy the Markov property. [ABSTRACT FROM AUTHOR]

It is shown that if a Quantum Markov semigroup is "multiplicatively" perturbed under suitable conditions, the resulting minimal semigroup remains Markov (or conservative). [ABSTRACT FROM AUTHOR]

We study the optimal martingale transport problem under an additional constraint imposing the underlying process to be Markovian. This formulation results in a modified transportation problem in which the solutions correspond to robust price bounds for exotic derivatives within the class of calibrated martingale models exhibiting the Markov property. We investigate the arising consequences which comprise a dual perspective of the transport problem in terms of liquid replication strategies. Eventually an empirical investigation illustrates the influence of the Markov property on robust price bounds for financial derivatives. [ABSTRACT FROM AUTHOR]

Let X(t), t ∈ ℝ , be a d-dimensional vector-valued Brownian motion, d ≥ 1. For all b ∈ ℝ d ∖ (− ∞ , 0 ] d we derive exact asymptotics of ℙ { X (t + s) − X (t) > u b for some t ∈ [ 0 , T ] , s ∈ [ 0 , 1 ] } as u → ∞ , that is the asymptotical behavior of tail distribution of vector-valued analog of Shepp-statistics for X; we cover not only the case of a fixed time-horizon T > 0 but also cases where T → 0 or T → ∞ . Results for high level excursion probabilities of vector-valued processes are rare in the literature, with currently no available approach suitable for our problem. Our proof exploits some distributional properties of vector-valued Brownian motion, and results from quadratic programming problems. As a by-product we derive a new inequality for the 'supremum' of vector-valued Brownian motions. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the stochastic properties of some large size (area) polyominoes' perimeter such that the directed column-convex polyomino, the columnconvex polyomino, the directed diagonally-convex polyomino, the staircase (or parallelogram) polyomino, the escalier polyomino, the wall (or bargraph) polyomino. All polyominoes considered here are made of contiguous, not-empty columns, without holes, such that each column must be adjacent to some cell of the previous column. We compute the asymptotic (for large size n) Gaussian distribution of the perimeter, including the corresponding Markov property of the chain of columns, and the convergence to classical Brownian motions of the perimeter seen as a trajectory according to the successive columns. All polyominoes of size n are considered as equiprobable. [ABSTRACT FROM AUTHOR]