Limiting behavior of generalized delayed average of random sequence

Let (ξ _n ) _n∈N be a sequence of arbitrarily dependent random variables, and let (T_{a_n ,k_n }) be delayed sums. By virtue of the notion of asymptotic delayed log-likelihood ratio and Laplace transformation, the almost sure limiting behavior of the generalized delayed averages (frac{1} {{k_n }}T_{a_n ,k_n }) is investigated, of which the results generalize and improve some work of Liu.