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Concept# Random field

Résumé

In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb{R}^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb{R}^n(or some other domain). It is also sometimes thought of as a synonym for a stochastic process with some restriction on its index set. That is, by modern definitions, a random field is a generalization of a stochastic process where the underlying parameter need no longer be real or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold.
Formal definition
Given a probability space (\Omega, \mathcal{F}, P), an X-valued random field is a collection of X-valued random variables indexed by elements in a topological space T. That is, a random field F is a collection
: { F_t : t \in T }
where each F_t is an X-va

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Introduction to the basic techniques of image processing. Introduction to the development of image-processing software and to prototyping in JAVA. Application to real-world examples in industrial vision and biomedical imaging.

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An accurate assessment of the risk of extreme environmental events is of great importance for populations, authorities and the banking/insurance/reinsurance industry. Koch (2017) introduced a notion of spatial risk measure and a corresponding set of axioms which are well suited to analyze the risk due to events having a spatial extent, precisely such as environmental phenomena. The axiom of asymptotic spatial homogeneity is of particular interest since it allows one to quantify the rate of spatial diversification when the region under consideration becomes large. In this paper, we first investigate the general concepts of spatial risk measures and corresponding axioms further and thoroughly explain the usefulness of this theory for both actuarial science and practice. Second, in the case of a general cost field, we give sufficient conditions such that spatial risk measures associated with expectation, variance, value-at-risk as well as expected shortfall and induced by this cost field satisfy the axioms of asymptotic spatial homogeneity of order 0, −2, −1 and −1, respectively. Last but not least, in the case where the cost field is a function of a max-stable random field, we provide conditions on both the function and the max-stable field ensuring the latter properties. Max-stable random fields are relevant when assessing the risk of extreme events since they appear as a natural extension of multivariate extreme-value theory to the level of random fields. Overall, this paper improves our understanding of spatial risk measures as well as of their properties with respect to the space variable and generalizes many results obtained in Koch (2017).

2019Given a sequence L & x2d9;epsilon of Levy noises, we derive necessary and sufficient conditions in terms of their variances sigma 2(epsilon) such that the solution to the stochastic heat equation with noise sigma(epsilon)-1L & x2d9;epsilon converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive and multiplicative noise and hence lift the findings of Asmussen and Rosinski (J Appl Probab 38(2):482-493, 2001), Cohen and Rosinski (Bernoulli 13(1):195-210, 2007) for finite-dimensional Levy processes to the infinite-dimensional setting without making distributional assumptions on the solutions such as infinite divisibility. One important ingredient of our proof is to characterize the solution to the limit equation by a sequence of martingale problems. To this end, it is crucial to view the solution processes both as random fields and as cadlag processes with values in a Sobolev space of negative real order.

Clément Dombry, Erwan Fabrice Koch

Max-stable random fields are very appropriate for the statistical modelling of spatial extremes. Hence, integrals of functions of max-stable random fields over a given region can play a key role in the assessment of the risk of natural disasters, meaning that it is relevant to improve our understanding of their probabilistic behaviour. For this purpose, in this paper, we propose a general central limit theorem for functions of stationary max-stable random fields on R-d. Then, we show that appropriate functions of the Brown-Resnick random field with a power variogram and of the Smith random field satisfy the central limit theorem. Another strong motivation for our work lies in the fact that central limit theorems for random fields on R-d have been barely considered in the literature. As an application, we briefly show the usefulness of our results in a risk assessment context. (C) 2018 Elsevier B.V. All rights reserved.

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