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Glasnik matematički, Vol. 58 No. 1, 2023.

Original scientific paper

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

A new definition of random set

Vesna Gotovac DJogaš ; Department of Mathematics, Faculty of Science, University of Split, 21000 Split, Croatia
Kateřina Helisová ; Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague 6, Czech Republic
Lev B Klebanov ; Department of Probability and Mathematical Statistics, Charles University, 18675 Prague 8, Czech Republic
Jakub Staněk orcid id orcid.org/0000-0003-1987-427X ; Department of Mathematics Education, Charles University, 18675 Prague 8, Czech Republic
Irina V Volchenkova ; Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague 6, Czech Republic

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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.

Keywords

Boolean algebra, Hilbert space isometry, measurable space, negative definite kernel, symmetric difference

Hrčak ID:

304397

URI

https://hrcak.srce.hr/304397

Publication date:

20.6.2023.

Visits: 563 *

Publication date: 30 June 2023

Volume: Vol 58

Issue: Svezak 1

Pages: 135-154

DOI: 10.3336/gm.58.1.10

Article Information (continued)

Categories:

Subject: Izvorni znanstveni članak

Keywords:

Keyword: Boolean algebra, Hilbert space isometry, measurable space, negative definite kernel, symmetric difference

A new definition of random set

Vesna Gotovac DJogaš[1]

Email: vgotovac@pmfst.hr

Kateřina Helisová[2]

Email: helisova@math.feld.cvut.cz

Lev B Klebanov[3]

Email: lev.klebanov@mff.cuni.cz

Jakub Staněk[4]

Email: stanekj@karlin.mff.cuni.cz

Irina V Volchenkova[5]

Email: i.v.volchenkova@gmail.com

 

[1] Department of Mathematics, Faculty of Science, University of Split, 21000 Split, Croatia

[2] Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague 6, Czech Republic

[3] Department of Probability and Mathematical Statistics, Charles University, 18675 Prague 8, Czech Republic

[4] Department of Mathematics Education, Charles University, 18675 Prague 8, Czech Republic

[5] Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague 6, Czech Republic

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.

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