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Probability Bounds for Product Poisson Processes

  • Joong Sung Kwon
  • Ronald Pyke
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 114)

Abstract

Consider processes formed as products of independent Poisson and symmetrized Poisson processes. This paper provides exponential bounds for the tail probabilities of statistics representable as integrals of bounded functions with respect to such product processes. In the derivations, tail probability bounds are also obtained for product empirical measures. Such processes arise in tests of independence. A generalization of the Hanson-Wright inequality for quadratic forms is used in the symmetric case. The paper also provides some reasonably tractable approximations to the more general bounds that are derived first.

Keywords

Brownian Motion Quadratic Form Poisson Process Bond Price Compound Poisson Process 
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References

  1. Alder, R.J. and Feigin, P.D. (1984), On the cadlaguity of random measures. Ann. Prob. 12, 615–630.CrossRefGoogle Scholar
  2. Arcones, M.A. and Giné, E. (1994), Limit theorems for U-processes. Ann. Prob. 21, 1494–1542.CrossRefGoogle Scholar
  3. Bachelier, L., Théorie de la speculation, Ann.Sci. École Norm. Sup. 3, 21–86 (English translation by A. J. Boness: inThe Random Character of Stock Market Prices17–78, P. H. Cootner, Editor, M.I.T. Press, Cambridge, Mass. (1967).Google Scholar
  4. Bass, R.F. and Pyke, R. (1984), The existence of set-indexed Lévy processes. Z. Wahrsch. verw. Gebiete. 66, 157–172.MathSciNetMATHCrossRefGoogle Scholar
  5. Bass, R.F. and Pyke, R. (1987), A central limit theorem for D(A) valued processes. Stochastic Processes and their Applications. 24, 109–131.MathSciNetMATHCrossRefGoogle Scholar
  6. Bennett, G. (1962), Probability inequalities for the sums of bounded random variables. J.A.S.A. 57, 33–45.MATHGoogle Scholar
  7. Einstein, A. (1905). On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat. Ann. Phys. 17, 549–560.MATHCrossRefGoogle Scholar
  8. Hanson, D.L. and Wright, F.T. (1971), A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Statist. 42, 1079–1083.MathSciNetMATHCrossRefGoogle Scholar
  9. Hoeffding, W. (1963), Probability inequalities for sums of bounded random variables. J.A.S.A. 57, 33–45.Google Scholar
  10. Jain, N. and Marcus, M. (1978), Continuity of subgaussian processes. In Probability on Banach Spaces 81–196. Edited by J. Kuelbs. Marcel Dekker, NY.Google Scholar
  11. Ossiander, M. (1991), Product measure and partial sums of Bernoulli random variables. In preparation. 158 Joong Sung Kwon, Ronald PykeGoogle Scholar
  12. Ossiander, M. (1992), Bernstein inequalities for U-statistics. To appear.Google Scholar
  13. Pyke, R. (1983), A uniform central limit theorem for partial-sum processes indexed by sets. Probab. Statist. and Anal. (Edited by J. F. C. Kingman and G. E. H. Reuter) London Math. Soc. Lect. Notes. Ser. 79, 219–240.Google Scholar
  14. Pyke, R. (1996). The early history of Brownian motion. In preparation.Google Scholar
  15. Rao, C. R. (1973) Linear Statistical Inference and its Applications. (2nd Edition). J. Wiley and Sons, New York.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Joong Sung Kwon
    • 1
  • Ronald Pyke
    • 2
  1. 1.Department of MathematicsSun Moon UniversityAsan, ChungnamSouth Korea
  2. 2.Department of MathematicsUniversity of washingtonSeattleUSA

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