Poisson approximation of processes with locally independent increments and Markov switching

Limnios, N.; Samoilenko, I.

doi:10.1090/S0094-9000-2015-00939-X

Poisson approximation of processes with locally independent increments and Markov switching

Authors: N. Limnios and I. V. Samoilenko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 89 (2013).
Journal: Theor. Probability and Math. Statist. 89 (2014), 115-126
MSC (2010): Primary 60J55, 60B10, 60F17, 60K10; Secondary 60G46, 60G60
DOI: https://doi.org/10.1090/S0094-9000-2015-00939-X
Published electronically: January 26, 2015
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Abstract: The weak convergence of additive functionals is studied for stochastic processes with locally independent increments and with Markov switching in the scheme of Poisson approximation. The singular perturbation problem for the generator of a Markov process is used to prove the relative compactness.

1. J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564 (98e:60117)
2. E. Çinlar, J. Jacod, P. Protter, and M. J. Sharpe, Semimartingale and Markov processes, Z. Wahrschein. verw. Gebiete 54 (1980), 161-219. MR 597337 (82h:60084)
3. C. Dellacherie, Capacités et processus stochastiques, Springer-Verlag, Berlin-Heidelberg-New York, 1972. MR 0448504 (56:6810)
4. S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, J. Wiley, New York, 1986. MR 838085 (88a:60130)
5. I. I. Gihman and A. V. Skorohod, Theory of Stochastic Processes, vol. 1-3, Springer, Berlin, 1974. MR 0346882 (49:11603)
6. J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987. MR 959133 (89k:60044)
7. V. S. Koroliuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific Publishers, Singapore, 2005. MR 2205562 (2007a:60004)
8. V. S. Koroliuk, N. Limnios, and I. V. Samoilenko, Poisson approximation of processes with locally independent increments with Markov switching, Theory Stoch. Process. 15(31) (2009), no. 1, 40-48; corrections in Letter from the publishers, Theory Stoch. Process. 15(31) (2009), no. 2, 164. MR 2598522 (2010m:60112)
9. H. J. Kushner, Weak Convergence Methods and Singular Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, 1990. MR 1102242 (92d:93003)
10. R. Sh. Liptser, The Bogolyubov averaging principle for semimartingales, Proceedings of the Steklov Institute of Mathematics (1994), no. 4. MR 1383305 (97c:60117)
11. R. Sh. Liptser and A. N. Shiryayev, Theory of Martingales, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1989. MR 1022664 (90j:60046)
12. I. V. Samoilenko, Large deviations for random evolutions with independent increments in the scheme of the Poisson approximation, Theor. Probab. Math. Statist. 85 (2012), 107-114. MR 2933707
13. K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. MR 1739520 (2003b:60064)
14. D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, 1979. MR 532498 (81f:60108)
15. A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, AMS, Providence, RI, 1989.MR 1020057
16. A. V. Skorokhod, F. C. Hoppensteadt, and H. Salehi, Random Perturbation Method with Application in Science and Engineering, Springer, New York, 2002. MR 1912425 (2003m:34129)
17. D. S. Silvestrov, Limit Theorems for Randomly Stopped Stochastic Processes, Probability and its Applications, Springer, Berlin 2004. MR 2030998 (2005b:60003)
18. M. N. Sviridenko, Martingale approach to limit theorems for semi-Markov processes, Theor. Probab. Appl. (1986), 540-545. MR 1022647 (91b:60072)