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Unconditional Glivenko-Cantelli-type theorems and weak laws of large numbers for bootstrap

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  • Arenal-Gutiérrez, Eusebio
  • Matrán, Carlos
  • Cuesta-Albertos, Juan A.

Abstract

We first give some Glivenko-Cantelli-type theorems for the bootstrap. We prove almost sure convergence and convergence in probability. The results involve conditions on the resampling size which are not far from being (and often are) optimal. We next analyze the weak law of large numbers for the bootstrap mean. The conditions needed for these weak laws are related to the Glivenko-Cantelli statements on convergence in probability. This relation makes it possible to obtain weak laws in very general situations. In particular our techniques work with smoothed boostrap, and even lead to a proof of the strong law for the mean when kernel estimators are used.

Suggested Citation

  • Arenal-Gutiérrez, Eusebio & Matrán, Carlos & Cuesta-Albertos, Juan A., 1996. "Unconditional Glivenko-Cantelli-type theorems and weak laws of large numbers for bootstrap," Statistics & Probability Letters, Elsevier, vol. 26(4), pages 365-375, March.
    • Handle: RePEc:eee:stapro:v:26:y:1996:i:4:p:365-375
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    References listed on IDEAS

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    1. Burke, Murray D. & Gombay, Edit, 1988. "On goodness-of-fit and the bootstrap," Statistics & Probability Letters, Elsevier, vol. 6(5), pages 287-293, April.
    2. Wellner, Jon A., 1981. "A Glivenko-Cantelli theorem for empirical measures of independent but non-identically distributed random variables," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 309-312, August.
    3. Galen R. Shorack, 1979. "The weighted empirical process of row independent random variables with arbitrary distribution functions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 33(4), pages 169-189, December.
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    Cited by:

    1. Einmahl, J.H.J. & Rosalsky, A., 2004. "General Weak Laws of Large Numbers for Bootstrap Sample Means," Other publications TiSEM 8ccbe78d-31bd-4641-b50b-c, Tilburg University, School of Economics and Management.
    2. Mojirsheibani, Majid, 2001. "The Glivenko-Cantelli theorem based on data with randomly imputed missing values," Statistics & Probability Letters, Elsevier, vol. 55(4), pages 385-396, December.
    3. Rosalsky, Andrew & Sreehari, M., 1998. "On the limiting behavior of randomly weighted partial sums," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 403-410, November.

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