Summary
We investigate the ergodic properties of spatial processes, i.e. stochastic processes with an index set of bounded Borel subsets in ℝv, and prove mean and individual ergodic theorems for them. As important consequences we get a generalization of McMillan's theorem due to Fritz [4]; the existence of specific energy for a large class of interactions in the case of marked point processes in ℝv and the existence of the specific Minkowski Quermaßintegrals for Boolean models in ℝv with convex, compact grains.
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Behmer, K.-H.: Zufällige Schnitte. Diplomarbeit, Bielefeld (1976)
Csiszár, I.: On generalized entropy. Studia Sci. Math. Hungar.4, 401–419 (1969)
Davy, P.: Projected thick sections through multi-dimensional particle aggregates. J. Appl. Probability13, 714–722 (1977)
Fritz, J.: Generalization of McMillan's theorem to random set functions. Studia Sci. Math. Hungar.5, 369–394 (1970)
Föllmer, H.: On entropy and information gain in random fields. Z. Wahrscheinlichkeitstheorie verw. Gebiete26, 207–217 (1973)
Gallavotti, G., Miracle-Solé, S.: A variational principle for the equilibrium of hard sphere systems. Ann. Inst. H. Poincaré VIII, 287–299 (1968)
Ionescu Tulcea, A.: Contributions to information theory for abstract alphabets. Arkiv Math.4, 235–247 (1963)
Jacobs, K.: Lecture Notes on Ergodic Theory I, II (1962/63)
Kingman, J.C.: The ergodic theory of subadditive processes, J. Roy. Statist. Soc. Ser. B30, 499–510 (1968)
Krickeberg, K.: Stochastische Konvergenz von Semimartingalen. Math. Z.66, 470–486 (1957)
Krickeberg, K.: Processus spatiaux. Lectures Universities Paris V, VI, XI (1975/76)
Krickeberg, K.: Statistics of point processes. Invited lecture: 7th conference on stochastic processes and their applications, Enschede, The Netherlands (1977)
Lebowitz, J.L., Presutti, E.: Statistical mechanics of systems of unbounded spins. Comm. Math. Phys.50, 195–218 (1976)
Maker, P.: The ergodic theorem for a sequence of functions. Duke Math. J.6, 27–20 (1940)
Matheron, G.: Randoms sets & Integral Geometry. New York-London-Sydney-Toronto: Wiley 1975
Miles, R.E.: On the homogeneous planar Poisson point process. Math. Biosci.6, 85–127 (1970)
Minlos, R.A.: The regularity of the Gibbs limiting distribution. Funkcional Anal. i. Prilozen1, 40–45 (1967)
Minlos, R.A.: Lectures on Statistical Physics. Russian Math. Surveys23, 137–194 (1968)
Neveu, J.: Mathematical foundations of the calculus of probability. San Francisco, London, Amsterdam: Holden-Day 1965
Nguyen, X.X., Zessin, H.: Punktprozesse mit Wechselwirkung. Z. Wahrscheinlichkeitstheorie verw. Gebiete37, 91–126 (1976)
Nguyen, X.X.: Ergodic theorems for subaddive spatial processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete48, 159–176 (1979)
Nguyen, X.X., Zessin, H.: Integral and differential characterization of the Gibbs process. Math. Nachr. (to appear)
Perez, A.: Notions généralisées d'incertitude, d'entropie et d'information du point de vue de la theorie des martingale, Trans 1st Pragne Conference on Information Theory (1957)
Ruelle, D.: Statistical mechanics. New York, Amsterdam: W.A. Benjamin 1969
Ruelle, D.: Supertable interaction in classical statistical mechanics. Comm. Math. Phys.18, 127–159 (1970)
Ruelle, D.: Probability estimates for continuous spin systems. Comm. Math. Phys.50, 189–194 (1976)
Smythe, R.T.: Multiparameter subadditive processes. Ann. Probability4, 772–782 (1976)
Tempel'man, A.A.: Ergodic theorems for general dynamical system. Trans. Moscow Math. Soc.26, 94–132 (1972)
Thouvenot, J.P.: Convergence en moyenne de l'information pour l'action de Z2. Z. Wahrscheinlichkeitstheorie verw. Gebiete24, 135–137 (1972)
Widom, B., Rowlinson, J.S.: New model for the study of liquid-vapor phase transitions. J. Chem. Phys.52, 1670–1684 (1970)
Jolivet, E.: Caracterisation et test du caractere agregatif des processus ponctuels stationaires sur ℝ2. In: Journées de Statistique des Processus Stochastique, Lecture Notes in Mathematics636, Berlin-Heidelberg-New York: Springer 1968
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Dedicated to Klaus Krickeberg on the occasion of his 50th birthday
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Nguyen, X.X., Zessin, H. Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 48, 133–158 (1979). https://doi.org/10.1007/BF01886869
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DOI: https://doi.org/10.1007/BF01886869