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Random sets and their asymptotic measure. (English) Zbl 0773.60010
Summary: The paper continues the investigation begun by [J. Hurt, J. Machek, J. Štěpán and D. Vorlíčková, ibid. 32, 229-237 (1982; Zbl 0497.60013)]and the authors [Information theory, statistical decision functions, random processes, Trans. 10th Prague Conf., Prague/Czech. 1986, Vol. B, 349-356 (1989; Zbl 0688.60008)] by presenting a probabilistic model for random sets suitable to handle the weak convergence of processes given by $$\text{Meas}[X_ n\cap I(t)]- t \text{Meas}(X_ n)$$, $$t\in[0,1]$$, where $$X_ n$$ are random sets and $$I(t)$$ is a non-random set valued process indexed by [0,1]. Moreover, sufficient conditions are found to ensure the asymptotic normality of r.v.’s $$\text{Meas}[X_ n\cap Y_ n]-\text{Meas}[X_ n]\cdot\text{Meas}[Y_ n]$$, where $$X_ n$$ and $$Y_ n$$ are independent random sets.
##### MSC:
 60D05 Geometric probability and stochastic geometry 60F17 Functional limit theorems; invariance principles
##### Keywords:
weak convergence; asymptotic normality; random sets
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##### References:
 [1] BILLINGSLEY P.: Convergence of Probability Measures. J. Wiley, New York, 1968. · Zbl 0172.21201 [2] DENI J., CHOQUET G.: Sur l’equation de convolution \mu = \mu * \sigma . C. R. Acad. Sci. Paris Sér. I Math. 250 (1960), 799-801. · Zbl 0093.12802 [3] HALMOS P. R.: Lectures on Ergodic Theory. (Russian Translation), Izd. In. Lit., Moscow, 1959. · Zbl 0073.09302 [4] HALMOS P. R.: Measure Theory. Van Nostrand, London, 1968. [5] HURT J., MACHEK J., ŠTĚPÁN J., VORLÍČKOVÁ D.: The intersections of random finite sets. Math. Slovaca 32 (1982), 229-237. · Zbl 0497.60013 · eudml:32386 [6] STRAKA F.: Random Sets and their Intersections. (Czech), PhD-theses, Charles University, Prague, 1986. · Zbl 0643.60010 [7] STRAKA F., ŠTĚPÁN J.: Random sets in [0, 1]. Proc. of 10th Prague Conference on Information Theory 1986, Academia, Prague, 1988, pp. 349-355. [8] SCHWARTZ L.: Radon Measures. Oxford University Press, Oxford, 1973. · Zbl 0298.28001 [9] ŠTĚPÁN J.: Some notes on the convolution semigroup of probabilities on a metric group. Comment. Math. Univ. Carolin. 10 (1969), 613-623. · Zbl 0193.44702 · eudml:16347
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