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On the tail behaviour of the distribution function of multiple stochastic integrals
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  • Published: July 1988

On the tail behaviour of the distribution function of multiple stochastic integrals

  • P. Major1 

Probability Theory and Related Fields volume 78, pages 419–435 (1988)Cite this article

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  • 6 Citations

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Summary

Let F n (u) denote the empirical distribution function of a sample of i.i.d. random variables with uniform distribution on [0, 1]. Define \(\bar \mu _n^* (u) = \sqrt n [F_n (u) - u]\), and consider the integrals \(I(t) = \int\limits_0^t {\int\limits_0^1 {...\int\limits_0^1 {f(u_1 ,...,u_s )} } }\) where f is a bounded measurable function. We give a good upper bound on the probability \(P{\text{ }}(\mathop {\sup }\limits_{0 \leqq t \leqq 1} {\text{ | }}I{\text{(}}t{\text{) |}} \geqq x)\). An analogous estimate is given for multiple integrals with respect to a Poisson process.

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References

  1. Major, P., Rejtö, L.: Strong embedding of estimation of the distribution function under random censorship. Ann. Statist., 15 (1988)

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Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, Reáltanoda u. 13-15, P.O.B. 127, H-1364, Budapest, V, Hungary

    P. Major

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  1. P. Major
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Cite this article

Major, P. On the tail behaviour of the distribution function of multiple stochastic integrals. Probab. Th. Rel. Fields 78, 419–435 (1988). https://doi.org/10.1007/BF00334204

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  • Received: 25 April 1986

  • Revised: 13 January 1988

  • Issue Date: July 1988

  • DOI: https://doi.org/10.1007/BF00334204

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Keywords

  • Distribution Function
  • Uniform Distribution
  • Stochastic Process
  • Probability Theory
  • Measurable Function
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