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Limit theorems for jump shock models

Published online by Cambridge University Press:  14 July 2016

Keigo Yamada
Keigo Yamada*
Affiliation:
University of Tsukuba
*
∗ Postal address: Institute of Information Sciences and Electronics, University of Tsukuba, Ibaraki 305, Japan.
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Abstract

We consider an additive shock process where shocks occur according to a Poisson point process and they are accumulated in an appropriate way to the damage. It is shown that suitably normalized shock processes converge weakly to a process which is represented as a sum of a stable process and a deterministic process.

Keywords

STABLE PROCESSPOISSON POINT PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 4 , December 1989 , pp. 793 - 806
Copyright
Copyright © Applied Probability Trust 1989 

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References

[1] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Breiman, L. (1968) Probability. Addison-Wesley Reading, Mass. Google Scholar
[3] Drosen, J. W. (1986) Pure jump shock models in reliability. Adv. Appl. Prob. 18, 423–440.CrossRefGoogle Scholar
[4] Ikeda, N. and Watanabe, S. (1981) Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
[5] Jacod, J. (1985) Theorems limit pour les processus. In École d'Èté de Probabilités de Saint-Flour XIII-1983, Lecture Notes in Mathematics 1117, Springer-Verlag, Berlin.Google Scholar
[6] Jacod, J. and Shiryaev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[7] Kasahara, Y. and Maejima, M. (1986) Functional limit theorems for weighted sums of i.i.d. random variables. Prob. Theory Rel. Fields 72, 161–183.CrossRefGoogle Scholar
[8] Kasahara, Y. and Maejima, M. (1988) Weighted sums of i.i.d. random variables attracted to integrals of stable processes. Prob. Theory Rel. Fields 78, 75–96.CrossRefGoogle Scholar
[9] Kasahara, Y. and Watanabe, S. (1986) Limit thoerems for point processes and their functionals. J. Math. Soc. Japan 38, 543–574.CrossRefGoogle Scholar
[10] Lenglart, E. (1977) Relation de domination entre deux processus. Ann. Inst. H. Poincaré B XIII, 171–179.Google Scholar
[11] Lindvall, T. (1973) Weak convergence of probability measures and random functions in the function space D[0, 8). J. Appl. Prob. 10, 109–121.CrossRefGoogle Scholar
[12] Liptser, R. S. and Shiryaev, A. N. (1980) A functional central limit theorem for semimartingales. Theory Prob. Appl. 25, 667–689.CrossRefGoogle Scholar