Large deviation principles for sequences of logarithmically weighted means

In this paper we consider several examples of sequences of partial sums of triangular arrays of random variables { X n : n ⩾ 1 } ; in each case X n 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 { 1 log n ∑ k = 1 n 1 k X k : n ⩾ 1 } with speed function v n = log n . We also prove a sample path large deviation principle for { X n : n ⩾ 1 } defined by X n ( ⋅ ) = ∑ i = 1 n U i ( σ 2 ⋅ ) n , where σ 2 ∈ ( 0 , ∞ ) and { U n : n ⩾ 1 } is a sequence of independent standard Brownian motions.