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An inequality for a metric in a random collision process

Published online by Cambridge University Press:  14 July 2016

Shōichi Nishimura
Shōichi Nishimura*
Affiliation:
Tokyo Institute of Technology
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Abstract

A random collision process with transition probabilities belonging to the same type of distribution is considered. It was proved that if the characteristic function of the initial distribution has a positive radius of convergence, then the sequence {Fn (x)} converges weakly to a distribution G(x) [9]. We define a metric e 1[Fn ;G], which is analogous to the functional e[f] introduced by Tanaka to Kac's model of Maxwellian gas [10]. We prove that e 1[Fn ; G] is monotone non-increasing as n → ∞, and also the convergence of the sequence {Fn (x)} under the weaker assumption that for some a > 1 the initial distribution has an ath moment.

Keywords

RANDOM COLLISION PROCESSKAC'S MODEL OF A MAXWELLIAN GASFUNCTIONAL eGLOBAL VERSION OF LIMIT THEOREM

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 2 , June 1975 , pp. 239 - 247
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Agnew, R. P. (1954) Global version of the central limit theorem. Proc. Nat. Acad. Sci. 40, 800–804.CrossRefGoogle ScholarPubMed
[2] Agnew, R. P. (1957) Estimates for global central limit theorem. Ann. Math. Statist. 28, 26–42.Google Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. 2. John Wiley, New York.Google Scholar
[4] Fréchet, M. (1957) Sur la distance de deux lois de probabilité. Publ. Inst. Statist. Univ. Paris. Vol. VI, Fasc. 3, 183–198.Google Scholar
[5] Lukacs, E. (1970) Characteristic Functions. Second edition. Griffin, London.Google Scholar
[6] Moran, P. A. P. (1961) Entropy, Markov processes and Boltzmann's H-theorem. Proc. Camb. Phil. Soc. 57, 833–842.Google Scholar
[7] Murata, H. and Tanaka, H. (1974) An inequality for certain functional of multidimensional probability distributions. Hiroshima Math. J. 4, 75–81.Google Scholar
[8] Nishimura, S. (1974) Random collision processes and their limiting distribution using the discrimination information J. Appl. Prob. 11, 266–280.Google Scholar
[9] Nishimura, S. (1974) Random collision processes with transition probability belonging to the same type of distribution. J. Appl. Prob. 11, 703–714.Google Scholar
[10] Tanaka, H. (1973) An inequality for a functional of probability distributions and its application to Kac's one-dimensional model of a Maxwellian gas. Z. Wahrscheinlichkeitsth. 27, 47–52.CrossRefGoogle Scholar