Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T13:08:16.706Z Has data issue: false hasContentIssue false
  • English
  • Français

On a probability problem arising in the theory of counters

Published online by Cambridge University Press:  24 October 2008

L. Takács
L. Takács
Affiliation:
Institute for Applied MathematicsHungarian Academy of SciencesBudapest, Hungary
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper contains some remarks on the problems treated by Hammersley(9). With the aid of a new and simple method the exact solutions of Hammersley's problems and the asymptotic behaviour of the solutions are found. The problems in question are: A Poisson process is transformed into a new process by a Type II counter, and this new process is transformed again by a Type I counter. The distribution of the number of events occurring in each transformed sequence is to be determined.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 52 , Issue 3 , July 1956 , pp. 488 - 498
Copyright
Copyright © Cambridge Philosophical Society 1956

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

REFERENCES

(1)Albert, G. E. and Nelson, L.Contributions to the statistical theory of counter data. Ann. math. Statist. 24 (1953), 9–22.CrossRefGoogle Scholar
(2)Blackwell, D.A renewal theorem. Duke math. J. 15 (1948), 145–51.CrossRefGoogle Scholar
(3)Doob, J. L.Renewal theory from the point of view of the theory of probability. Trans. Amer. math. Soc. 63 (1948), 422–38.CrossRefGoogle Scholar
(4)Feller, W.On the integral equation of renewal theory. Ann. math. Statist. 12 (1941), 243–67.CrossRefGoogle Scholar
(5)Feller, W.On probability problems in the theory of counters. Courant anniversary volume (1948), pp. 105–15.Google Scholar
(6)Feller, W.Fluctuation theory of recurrent events. Trans. Amer. math. Soc. 67 (1949), 98–119.CrossRefGoogle Scholar
(7)Giltay, J.A counter arrangement with constant resolving time. Physica, 's Grav., 10 (1943), 725–34.Google Scholar
(8)Gnedenko, B. V.K teorii schotchikov Geiger-Müllera. Zh. exp. tear. Fiz. 11 (1941), 101–6.Google Scholar
(9)Hammersley, J. M.On counters with random dead time. I. Proc. Camb. phil. Soc. 49 (1953), 623–37.CrossRefGoogle Scholar
(10)Jost, Res.Bemerkungen zur mathematischen Theorie der Zähler. Helv. phys. acta, 20 (1947), 173–82.Google Scholar
(11)Kolmogorov, A. N. and Prohorov, Ju. V.O summah sluchainovo chisla sluchainyh slagaemyh. Usp. mat. Nauk, 4 (1949), 168–2.Google Scholar
(12)Kosten, L.On the frequency distribution of the Geiger-Müller counter in a constant interval. Physica, 's Grav., 10 (1940), 749–56.Google Scholar
(13)Kurbatov, J. D. and Mann, H. B.Correction of G-M counter data. Phys. Rev. (2), 68 (1945), 40–3.CrossRefGoogle Scholar
(14)Levert, C. and Scheen, W. L.Probability fluctuation of discharges in a Geiger-Müller counter produced by cosmic radiation. Physica, 's Grav., 10 (1943), 225–38.Google Scholar
(15)Malmquist, S.A statistical problem connected with the counting of radioactive particles. Ann. math. Statist. 18 (1947), 225–64.CrossRefGoogle Scholar
(16)Pollaczek, F.Sur la théorie stochastique des compteurs. C.R. Acad. Sci., Paris, 238 (1954), 766–7.Google Scholar
(17)Schiff, L. I.Statistical analysis of counter data. Phys. Rev. (2), 50 (1936), 88–96.CrossRefGoogle Scholar
(18)Takács, L.Wahrscheinlichkeitstheoretische Behandlung von Koinzidenz-Erscheinungen, mit Ereignissen gleicher Zeitdauer. Comptes rendus du prémier congrés des mathématiciens hongrois, 1950, pp. 731–40.Google Scholar
(19)Takács, L.Occurrence and coincidence phenomena in case of happenings with arbitrary distribution law of duration. Acta math. hung. 2 (1951), 275–98.Google Scholar
(20)Takács, L.A new method for discussing recurrent stochastic processes. Publications de l'Institut des Mathématiques Appliquées de l'Académic des Sciences de Hongrie, 2 (1953), 135–51 (Hungarian, English summary).Google Scholar
(21)Takács, L.On processes of happenings generated by means of a Poisson process. Acta math. hung. 6 (1955), 81–99.Google Scholar
(22)Takács, L. On stochastic processes in the theory of counters (to appear).Google Scholar
(23)Täcklind, S.Fourieranalytische Behandlung vom Erneuerungsproblem. Skand. Aktuar-Tidskr. (1945), 68–105.Google Scholar