Large deviation principles for sequences of logarithmically weighted means

Abstract

AbstractIn this paper we consider several examples of sequences of partial sums of triangular arrays of random variables {Xn:n⩾1}; in each case Xn converges weakly to an infinitely divisible distribution (a Poisson distribution or a centered Normal distribution). For each sequence we prove large deviation results for the logarithmically weighted means {1logn∑k=1n1kXk:n⩾1} with speed function vn=logn. We also prove a sample path large deviation principle for {Xn:n⩾1} defined by Xn(⋅)=∑i=1nUi(σ2⋅)n, where σ2∈(0,∞) and {Un:n⩾1} is a sequence of independent standard Brownian motions