Large deviations for weighted empirical mean with outliers

We study in this article the large deviations for the weighted empirical mean Ln=1n∑1nf(xin)⋅Zi, where (Zi)i∈N(Zi)i∈N is a sequence of RdRd-valued independent and identically distributed random variables with some exponential moments and where the deterministic weights f(xin) are m×dm×d matrices. Here f is a continuous application defined on a locally compact metric space (X,ρ)(X,ρ) and we assume that the empirical measure 1n∑i=1nδxin weakly converges to some probability distribution RR with compact support YY.The scope of this paper is to study the effect on the Large Deviation Principle (LDP) of outliers  , that is elements xi(n)n∈{xin,1≤i≤n} such that lim infn→∞ρ(xi(n)n,Y)>0. We show that outliers can have a dramatic impact on the rate function driving the LDP for LnLn. 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 Z1n. 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)].