Laws of large numbers for residual Cesà ro alpha-integrable sequences under dependence assumptions

Martingale difference, pairwise negative quadrant dependence and Lp-mixingale difference are three special kinds of dependence structures for random variables. By relating these dependence conditions to residual Cesàro alpha-integrability, as well as to strongly residual Cesàro alpha-integrability, Lp-convergence and strong laws of large numbers are studied, and significant improvements over earlier results are obtained.