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Theory of Probability and Mathematical Statistics

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Point processes subordinated to compound Poisson processes


Authors: K. V. Kobylych and L. M. Sakhno
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 94 (2016).
Journal: Theor. Probability and Math. Statist. 94 (2017), 89-96
MSC (2010): Primary 60G55, 60G50
DOI: https://doi.org/10.1090/tpms/1011
Published electronically: August 25, 2017
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Abstract: Point processes $ N^f (t)=N\bigl (H^f(t)\bigr )$, $ t>0$, are studied in the paper where $ N(t)$ is a Poisson process and $ H^f(t)$ is a subordinator with the Bernstein function $ f(\lambda )$. We present the probability distribution and moments of the first and second order of processes $ N^f(t)$ for the case where $ H^f(t)$ is a compound Poisson process with gamma distributed jumps. We also consider these processes with double and iterated time change.


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Additional Information

K. V. Kobylych
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 4E, Kyiv 03127, Ukraine
Email: kristina.kobilich@gmail.com

L. M. Sakhno
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 4E, Kyiv 03127, Ukraine
Email: lms@univ.kiev.ua

DOI: https://doi.org/10.1090/tpms/1011
Keywords: Point processes, Poisson processes, generalized Poisson processes, Bernstein function, subordinators
Received by editor(s): April 4, 2016
Published electronically: August 25, 2017
Article copyright: © Copyright 2017 American Mathematical Society