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Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

Part of: Stochastic processes Geometric probability and stochastic geometry Markov processes Inference from stochastic processes

Published online by Cambridge University Press:  23 June 2021

Jesper Møller  and
Eliza O’Reilly
Jesper Møller*
Affiliation:
Aalborg University
Eliza O’Reilly*
Affiliation:
California Institute of Technology
*
*Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Øst, Denmark.
**Postal address: Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Blvd. MC 305-16, Pasadena, CA 91125, USA. Email address: eoreilly@caltech.edu
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Abstract

For a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

Keywords

Ginibre point processisotropic determinantal point process on the sphereglobally most repulsive determinantal point processprojection kernelstationary determinantal point process in Euclidean space

MSC classification

Primary: 60G55: Point processes 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60D05: Geometric probability and stochastic geometry 62M30: Spatial processes

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 469 - 483
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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