Cesàro α-Integrability and Laws of Large Numbers-II

Abstract

For a sequence of integrable random variables, we introduce a new set of conditions called Cesaro α-Integrability and Strong Cesaro α-Integrability and show that, for α<1/2, these conditions that are strictly weaker than Cesaro Uniform Integrability and Strong Cesaro Uniform Integrability respectively, are sufficient for WLLN and SLLN to hold for a sequence of pairwise independent random variables. For some special kinds of dependent sequences of random variables also, Cesaro α-integrability for appropriate α is shown to be sufficient for WLLN to hold.