Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T08:16:59.367Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal convergence of multidimensional shot noise and rates of this convergence

Published online by Cambridge University Press:  01 July 2016

Lothar Heinrich  and
Volker Schmidt
Lothar Heinrich
Affiliation:
Mining Academy of Freiberg
Volker Schmidt*
Affiliation:
Mining Academy of Freiberg
*
∗Postal address: Bergakademie Freiberg, Sektion Mathematik, DDR-9200 Freiberg, Bernhard-von-Cotta-Str. 2, GDR.
Get access Rights & Permissions [Opens in a new window]

Abstract

Using a representation formula expressing the mixed cumulants of realvalued random variables by corresponding moments, sufficient conditions are given for the normal convergence of suitably standardized shot noise assuming that the generating stationary point process is independently marked and Brillinger mixing and that its intensity tends to ∞. Furthermore, estimates for the rate of this normal convergence are obtained by exploiting a general lemma on probabilities of large deviations and on the rate of normal convergence.

Keywords

HIGH DENSITYCUMULANTSMARKED POINT PROCESSESMIXING CONDITION
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 4 , December 1985 , pp. 709 - 730
Copyright
Copyright © Applied Probability Trust 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bickel, P. J. and Wichura, M. J. (1971) Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 41, 1656–1670.CrossRefGoogle Scholar
2. Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
3. Borovkov, A. A. (1967) On limit laws for service processes in multi-channel systems (in Russian). Sib. Matem. Zhurnal 8, 983–1004.Google Scholar
4. Borovkov, A. A. (1980) Asymptotic Methods in Queueing Theory (in Russian). Nauka, Moscow.Google Scholar
5. Brillinger, D. R. (1975) Statistical inference for stationary point processes. In Stochastic Processes and Related topics , ed. Puri, M. L., Academic Press, New York, 55–99.Google Scholar
6. Daley, D. J. (1971) The definition of a multi-dimensional generalization of shot noise. J. Appl. Prob. 8, 128–135.CrossRefGoogle Scholar
7. Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Wiley, Chichester.Google Scholar
8. Gnedenko, B. V., König, D. Et Al. (1983) Handbook of Queueing Theory (in German), Vol. I. Akademie-Verlag, Berlin.Google Scholar
9. Grandell, J. (1976) Doubly Stochastic Poisson Processes. Lecture Notes in Mathematics 529, Springer-Verlag, Berlin.CrossRefGoogle Scholar
10. Heinrich, L. (1983) On probabilities of large deviations for sums of random variables connected in a Markov chain not satisfying Cramer’s condition (in German). Liet. Mat. Rink. 23, 211–223.Google Scholar
11. Iglehart, D. L. (1965) Limit diffusion approximations for the many server queue and the repairman problem. J. Appl. Prob. 2, 429–441.CrossRefGoogle Scholar
12. Iglehart, D. L. (1973) Weak convergence of compound stochastic processes. Stoch. Proc. Appl. 1, 11–31.CrossRefGoogle Scholar
13. König, D., Matthes, K. and Nawrotzki, K. (1967) Generalizations of the Erlang and Engset Formulas (A Method in Queueing Theory) (in German). Akademie-Verlag, Berlin.Google Scholar
14. Leonov, V. P. and Shiryaev, A. N. (1959) Sur le calcul des semi-invariants. Teor. Verojatnost. i Primenen. 4, 342–355.Google Scholar
15. Loeve, M. (1977) Probability Theory , Vol. I. Springer-Verlag, New York.Google Scholar
16. Matches, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, Chichester.Google Scholar
17. Papoulis, A. (1965) Probability, Random Variables, and Stochastic Processes. McGrawHill, New York.Google Scholar
18. Papoulis, A. (1971) High density shot noise and Gaussianity. J. Appl. Prob. 8, 118–127.CrossRefGoogle Scholar
19. Rao, J. S. (1966) An application of stationary point processes to queueing theory and textile research. J. Appl. Prob. 3, 231–246.CrossRefGoogle Scholar
20. Rice, J. (1977). On generalized shot noise. Adv. Appl. Prob. 9, 553–565.CrossRefGoogle Scholar
21. Rudzkis, R., Saulis, L. and Statulevicius, V. (1978) A general lemma on probabilities of large deviations (in Russian). Liet. Mat. Rink. 18, 99–116.Google Scholar
22. Saulis, L. (1981) General lemmas on the approximation of the normal distribution (in Russian). Liet. Mat. Rink. 21, 175–189.Google Scholar
23. Schmidt, V. (1984) On shot noise processes induced by stationary marked point processes. J. Inform. Processing Cybernetics 20, 397–406.Google Scholar
24. Schmidt, V. (1985) Poisson bounds for moments of shot noise processes. Statistics 16, 253–262.CrossRefGoogle Scholar
25. Statulevicius, V. (1966) On large deviations. Z. Wahrscheinlichkeitsth. 6, 133–144.CrossRefGoogle Scholar
26. Statulevicius, V. A. (1970) On limit theorems for random functions I (in Russian). Liet. Mat. Rink. 10, 583–592.Google Scholar
27. Stone, C. J. (1966) On absolute continuous components and renewal theory. Ann. Math. Statist. 37, 271–275.CrossRefGoogle Scholar
28. Westcott, M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448–466.CrossRefGoogle Scholar
29. Westcott, M. (1976) On the existence of a generalized shot noise process. In, Studies in Probability and Statistics , ed. Williams, E. J., North-Holland, Amsterdam, 73–88.Google Scholar
30. Whitt, W. (1982) On the heavy-traffic limit theorem for GI/G/8 queues. Adv. Appl. Prob. 14, 171–190.CrossRefGoogle Scholar