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Large deviations for weighted empirical mean with outliers

Author

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  • Maïda, M.
  • Najim, J.
  • Péché, S.

Abstract

We study in this article the large deviations for the weighted empirical mean , where is a sequence of -valued independent and identically distributed random variables with some exponential moments and where the deterministic weights are mxd matrices. Here is a continuous application defined on a locally compact metric space and we assume that the empirical measure weakly converges to some probability distribution R with compact support . The scope of this paper is to study the effect on the Large Deviation Principle (LDP) of outliers, that is elements such that We show that outliers can have a dramatic impact on the rate function driving the LDP for Ln. We also show that the statement of a LDP in this case requires specific assumptions related to the large deviations of the single random variable . This is the main input with respect to a previous work by Najim [J. Najim, A Cramér type theorem for weighted random variables, Electron. J. Probab. 7 (4) (2002) 32 (electronic)].

Suggested Citation

  • Maïda, M. & Najim, J. & Péché, S., 2007. "Large deviations for weighted empirical mean with outliers," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1373-1403, October.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:10:p:1373-1403
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    References listed on IDEAS

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    1. Gamboa, F. & Rouault, A. & Zani, M., 1999. "A functional large deviations principle for quadratic forms of Gaussian stationary processes," Statistics & Probability Letters, Elsevier, vol. 43(3), pages 299-308, July.
    2. Bercu, B. & Gamboa, F. & Rouault, A., 1997. "Large deviations for quadratic forms of stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 75-90, October.
    3. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
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