Convergence of Series of Moments on General Exponential Inequality

For an array {Xn,j , 1 6 j 6 kn, n > 1} of random variables and a sequence {cn} of positive numbers, sufficient conditions are given under which, for all ε > 0, ∑∞ n=1 cnE [ max 16i6kn ∣∣ ∑i j=1(Xn,j − EXn,jI{|Xn,j|6δ}) ∣∣ − ε ]p + < ∞, where x+ denotes the positive part of x and p > 1, δ > 0. Our statements are announced in a general setting allowing to conclude the previous convergence for well-known dependent structures. As an application, we study complete consistency and consistency in the rth mean of cumulative sum type estimators of the change in the mean of dependent observations.