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Applying Dynkin’s isomorphism: an alternative approach to understand the Markov property of the de Wijs process. (English) Zbl 1339.60059

Summary: E. B. Dynkin’s seminal paper [Bull. Am. Math. Soc., New Ser. 3, 975–999 (1980; Zbl 0519.60046)] associates a multidimensional transient symmetric Markov process with a multidimensional Gaussian random field. This association, known as Dynkin’s isomorphism, has profoundly influenced the studies of Markov properties of generalized Gaussian random fields. Extending Dynkin’s isomorphism, we study here a particular generalized Gaussian Markov random field, namely, the de Wijs process that originated in Georges Matheron’s pioneering work on mining geostatistics and, following [P. McCullagh, Ann. Stat. 30, No. 5, 1225–1310 (2002; Zbl 1039.62003)], is now receiving renewed attention in spatial statistics. This extension of Dynkin’s theory associates the de Wijs process with the (recurrent) Brownian motion on the two-dimensional plane, grants us further insight into Matheron’s kriging formula for the de Wijs process and highlights previously unexplored relationships of the central Markov models in spatial statistics with Markov processes on the plane.

MSC:

60G60 Random fields
60J25 Continuous-time Markov processes on general state spaces
60G15 Gaussian processes
60J65 Brownian motion
60G50 Sums of independent random variables; random walks
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References:

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