Glasnik Matematicki, Vol. 58, No. 1 (2023), 135-154. \( \)

A NEW DEFINITION OF RANDOM SET

Vesna Gotovac Đogaš, Kateřina Helisová, Lev B. Klebanov, Jakub Staněk and Irina V. Volchenkova

Department of Mathematics, Faculty of Science, University of Split, 21000 Split, Croatia
e-mail:vgotovac@pmfst.hr

Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague 6, Czech Republic
e-mail:helisova@math.feld.cvut.cz

Department of Probability and Mathematical Statistics, Charles University, 18675 Prague 8, Czech Republic
e-mail:lev.klebanov@mff.cuni.cz

Department of Mathematics Education, Charles University, 18675 Prague 8, Czech Republic
e-mail:stanekj@karlin.mff.cuni.cz

Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague 6, Czech Republic
e-mail:i.v.volchenkova@gmail.com

Abstract.   A new definition of random sets is proposed in the presented paper. It is based on a special distance in a measurable space and uses negative definite kernels for continuation from the initial space to the one of the random sets. Motivation for introducing the new definition is that the classical approach deals with Hausdorff distance between realisations of the random sets, which is not satisfactory for statistical analysis in many cases. We place the realisations of the random sets in a complete Boolean algebra (B.A.) endowed with a positive finite measure intended to capture important characteristics of the realisations. A distance on B.A. is introduced as a square root of measure of symmetric difference between its two elements. The distance is then used to define a class of Borel subsets of B.A. Consequently, random sets are defined as measurable mappings taking values in the B.A. This approach enables us to use more general family of distances between realisations of random sets which allows us to make new statistical tests concerning equality of some characteristics of random set distributions. As an extra result, the notion of stability of newly defined random sets with respect to intersections is proposed and limit theorems are obtained.

2020 Mathematics Subject Classification.   60DXX, 62G10, 60F99, 06Exx

Key words and phrases.   Boolean algebra, Hilbert space isometry, measurable space, negative definite kernel, symmetric difference

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https://doi.org/10.3336/gm.58.1.10

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