Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T17:04:21.500Z Has data issue: false hasContentIssue false
  • English
  • Français

On the use of the characteristic functional in the analysis of some stochastic processes occurring in physics and biology

Published online by Cambridge University Press:  24 October 2008

M. S. Bartlett  and
David G. Kendall
M. S. Bartlett
Affiliation:
Department of MathematicsUniversity of Manchester
David G. Kendall
Affiliation:
Magdalen CollegeOxford
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. In a recent contribution to these Proceedings Alladi Rama-krishnan (23) has discussed a number of problems which arise when the development of a cascade shower of cosmic rays is considered from the standpoint of the theory of stochastic processes. As has often been remarked (see, for example, the important study by Niels Arley (1)), there is a close formal analogy between such physical phenomena and the growth of biological populations. In particular, if the distance of penetration (t) is identified with the time, and the energy (E) of a particle in the shower is replaced by the age (x) of an individual in the population, there emerges an obvious analogy between stochastic fluctuations in the energy spectrum of the shower and similar fluctuations in the age distribution of the population.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 1 , January 1951 , pp. 65 - 76
Copyright
Copyright © Cambridge Philosophical Society 1951

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)Arley, N.On the theory of stochastic processes and their application to the theory of cosmic radiation (Copenhagen, 1943).Google Scholar
(2)Bartlett, M. S.Stochastic processes (1946). (Notes of a course given at the University of North Carolina: there is a copy in the Library of the Royal Statistical Society.)Google Scholar
(3)Bartlett, M. S.On the theoretical specification and sampling properties of autocorrelated time-series. J. R. Statist. Soc., Suppl., 8 (1946), 27–41.Google Scholar
(4)Bartlett, M. S.Some evolutionary stochastic processes. J. R. Statist. Soc. (B), 11 (1949), 211–29.Google Scholar
(5)Bartlett, M. S.Recurrence times. Nature, London, 165 (1950), 727–8.CrossRefGoogle ScholarPubMed
(6)Bellman, R. and Harris, T. E.On the theory of age-dependent stochastic branching processes. Proc. Nat. Acad. Sci., Washington, 34 (1948), 601–4.CrossRefGoogle ScholarPubMed
(7)Bochner, S.Stochastic processes. Ann. Math. 48 (1947), 1014–61.CrossRefGoogle Scholar
(8)Doob, J. L.Time-series and harmonic analysis. Proc. Berkeley Symposium on Math. Statist, and Probability (1949), pp. 303–43.Google Scholar
(9)Feller, W.Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung. Acta Biotheoretica, 5 (1939), 11–40.CrossRefGoogle Scholar
(10)Feller, W.On the theory of stochastic processes, with particular reference to applications. Proc. Berkeley Symposium on Math. Statist. and Probability (1949), pp. 403–32.Google Scholar
(11)Feller, W.Fluctuation theory of recurrent events. Trans. American Math. Soc. 67 (1949), 98–119.CrossRefGoogle Scholar
(12)Furry, W. H.On fluctuation phenomena in the passage of high energy electrons through lead. Phys. Rev. 52 (1937), 569–81.CrossRefGoogle Scholar
(13)Kendall, D. G.On the generalized birth-and-death process. Ann. Math. Statist. 19 (1948), 1–15.CrossRefGoogle Scholar
(14)Kendall, D. G.On some modes of population growth leading to R. A. Fisher's logarithmic series distribution. Biometrika, 35 (1948), 6–15.CrossRefGoogle Scholar
(15)Kendall, D. G.On the role of variable generation time in the development of a stochastic birth process. Biometrika, 35 (1948), 316–30.CrossRefGoogle Scholar
(16)Kendall, D. G.Stochastic processes and population growth. J. R. Statist. Soc. (B), 11 (1949), 230–64.Google Scholar
(17)Kendall, D. G.Random fluctuations in the age-distribution of a population whose development is controlled by the simple birth-and-death process. J. R. Statist. Soc. (B), 12 (1950). (In the Press.)Google Scholar
(18)Krintchine, A.Mathematisches über die Erwartung vor einem öffentlichen Schalter. Matemat. Sbornik, 39 (1932), 73–84. (Russian: German summary.)Google Scholar
(19)Le Cam, L.Un instrument d'étude des fonctions aléatoires: la fonctionnelle caractéristique. C.R. Acad. Sci., Paris, 224 (1947), 710–11.Google Scholar
(20)Moyal, J. E.Stochastic processes and statistical physics. J. R. Statist. Soc. (B), 11 (1949), 150–210.Google Scholar
(21)Owen, A. R. G.The theory of genetical recombination. I. Proc. Roy. Soc. B, 136 (1949), 67–94.Google Scholar
(22)Palm, C.Intensitätsschwankungen im Fernsprechverkehr. Ericsson Technics, no. 44 (1943), pp. 1–189.Google Scholar
(23)Ramakrishnan, A.Stochastic processes relating to particles distributed in a continuous infinity of states. Proc. Cambridge Phil. Soc. 46 (1950), 596–602.CrossRefGoogle Scholar
(24)Rice, S. O.Mathematical analysis of random noise. Bell Syst. Tech. J. 23 (1944), 282–332, and 24 (1945), 46–156.CrossRefGoogle Scholar
(25)Yule, G. U.A mathematical theory of evolution, etc. Philos. Trans. B, 213 (1924), 21–87.Google Scholar
(26)Yule, G. U.On a method of investigating periodicities in disturbed series, with special reference to Wolfer's sunspot numbers. Philos. Trans. A, 226 (1927), 267–98.Google Scholar