Large deviations for weighted empirical mean with outliers

Abstract

We study in this article the large deviations for the weighted empirical mean L n = 1 n ∑ 1 n f ( x i n ) ⋅ Z i , where ( Z i ) i ∈ N is a sequence of R d -valued independent and identically distributed random variables with some exponential moments and where the deterministic weights f ( x i n ) are m × d matrices. Here f is a continuous application defined on a locally compact metric space ( X , ρ ) and we assume that the empirical measure 1 n ∑ i = 1 n δ x i n weakly converges to some probability distribution R with compact support Y . The scope of this paper is to study the effect on the Large Deviation Principle (LDP) of outliers, that is elements x i ( n ) n ∈ { x i n , 1 ≤ i ≤ n } such that lim inf n → ∞ ρ ( x i ( n ) n , Y ) > 0 . We show that outliers can have a dramatic impact on the rate function driving the LDP for L n . 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 Z 1 n . 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)].