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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
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References:
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