Almost sure convergence for weighted sums of φ-mixing random variables with applications

For a sequence {x _n , n ≥ 1} of φ-mixing random variables and a triangular array of constants {a _n ,k,1≤k≤n,n≥ 1} conditions are given to ensure that (Formula presented.) where {b _n } is a prescribed sequence of positive constants. Our statements are established at the extension of a Rosenthal type inequality for φ-mixing random variables and discard any assumption on the mixing coefficient. The proposed results are employed in well-known estimators arising from regression models in order to obtain its strong consistency.