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On weighted approximations in D[0, 1] with applications to self-normalized partial sum processes

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Abstract

Let X,X 1,X 2,
 be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S [nt], 0 ≩ t ≩ 1} where S n = Σ n j=1 X j , under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S [nt]=V n , 0 ≩ t ≩ 1}, where V 2 n = Σ n j=1 X 2 j . L p approximations of self-normalized partial sum processes are also discussed.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, Canada, K1S 5B6

    M. CsörgƑ & B. Szyszkowicz

  2. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    Q. Wang

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  1. M. CsörgƑ
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  2. B. Szyszkowicz
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  3. Q. Wang
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Correspondence to M. CsörgƑ.

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Dedicated to IstvĂĄn Berkes, SĂĄndor CsörgƑ and PĂ©ter Major in celebration of their sixtieth years

The research of M. CsörgƑ and B. Szyszkowicz is supported by their NSERC Canada Discovery Grants at Carleton University, Ottawa, and Q. Wang’s research is supported in part by Australian Research Council at University of Sydney.

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CsörgƑ, M., Szyszkowicz, B. & Wang, Q. On weighted approximations in D[0, 1] with applications to self-normalized partial sum processes. Acta Math Hung 121, 307–332 (2008). https://doi.org/10.1007/s10474-008-7216-5

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  • DOI: https://doi.org/10.1007/s10474-008-7216-5

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