Cesaro α-Integrability and laws of large numbers-II

For a sequence of random variables, a new set of properties called Cesaro alpha-Integrability and Strong Cesaro alpha-Integrability was recently introduced in an earlier paper and these properties were used to prove several new laws of large numbers, namely both Strong and Weak Laws of Large Numbers for pairwise-independent random variables as well as WLLN for some dependent sequences of random variables. In this paper, a set of weaker conditions called Residual Cesaro alpha-Integrability and Strong Residual Cesaro alpha-Integrability are introduced and significant improvements over earlier results are obtained. In addition, new results on L-p-convergence, for 0< p< 2, and SLLN for some dependent sequences are proved.