2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970

Archive

# Theory of Probability and Mathematical Statistics (Teoriya Imovirnostei ta Matematychna Statystyka)

## Point processes subordinated to compound Poisson processes

### K. V. Kobylych, L. M. Sakhno

Abstract: We investigate point processes $N^f (t)=N(H^f(t))$, $t>0$, where $N(t)$ is a Poisson process and $H^f(t)$ is the subordinator with Bern\u{s}tein function $f(\lambda)$. For the case when $H^f(t)$ is the compound Poisson process with gamma distributed jumps we present probabilistic distribution and moments of the first and second order of the processes $N^f(t)$, and also consider these processes with double and iterated time change.

Keywords: Point processes, Poisson processes, compound Poisson processes, Bernstein function,subordinators

Bibliography:
1. D.Applebaum, L$\acute{e}$vy Processes and Stochastic Calculus (second edition), Cambridge University Press.(2009).
2. L.Beghin, C. Macci, Alternative forms of compound fractional Poisson processes, Abstract and Applied Analysis, Hindawi Publishing Corporation, (2012).
3. R.Mendoza-Arriaga, V.Linetsky, Time-changed CIR default intensities with two-sided mean-reverting jumps, The Annals of Applied Probability, 24, no. 2: 811–856, (2014).
4. R.Garra, E.Orsingher, M.Scavino, Some probabilistic properties of fractional point processes.arXiv:1604.05235v1 (2016).
5. E. Orsingher and F. Polito. The space-fractional Poisson process, Statistics and Probability Letters, 82: 852–858, (2012).
6. E.Orsingher and B.Toaldo, Counting processes with Bernstein intertimes and random jumps, Journal of Applied Probability, 52: 1028–1044, (2015).
7. F.Polito, E.Scalas, Generalization of the Space-Fractional Poisson Process and its Connection to some L$\acute{e}$vy Processes, Electronic Communications in Probability, 21(20): 1–14, (2016).
8. K. Sato, L$\acute{e}$vy processes and infinitely divisible distributions, Cambridge University Press, (1999).