Large deviations for weighted empirical mean with outliers

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)].