Markov Random Fields
In this book we study Markov random functions of several variables. What is traditionally meant by the Markov property for a random process (a random function of one time variable) is connected to the concept of the phase state of the process and refers to the independence of the behavior of the process in the future from its behavior in the past, given knowledge of its state at the present moment. Extension to a generalized random process immediately raises nontrivial questions about the definition of a suitable" phase state," so that given the state, future behavior does not depend on past behavior. Attempts to translate the Markov property to random functions of multi-dimensional "time," where the role of "past" and "future" are taken by arbitrary complementary regions in an appro priate multi-dimensional time domain have, until comparatively recently, been carried out only in the framework of isolated examples. How the Markov property should be formulated for generalized random functions of several variables is the principal question in this book. We think that it has been substantially answered by recent results establishing the Markov property for a whole collection of different classes of random functions. These results are interesting for their applications as well as for the theory. In establishing them, we found it useful to introduce a general probability model which we have called a random field. In this book we investigate random fields on continuous time domains. Contents CHAPTER 1 General Facts About Probability Distributions §1.
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a-algebra of events absolutely continuous algebra arbitrary biorthogonal function Borel boundary conditions bounded complements bounded domain closed linear span closed set closure conditional distributions consider continuous linear converges corresponding countable covariance function defined denote derivatives differential form disjoint domains S C T domains S1 equivalent finite dimensional Fourier transform function 4.1 function v e V(T Hermite polynomials Hilbert space integrable intersection L-Markov Lemma linear functional Markov extension Markov property Markov with respect measurable with respect norm open domains probability measure probability space process H(t random field H(S random function random process random variables S G T satisfies semi-ring sequence space H space L*(Q space V(T spectral density f(A splits square-integrable stationary function stationary process stochastic subspace support Supp Theorem u e C3 u e Cô u e Cô(T