Cesàro α-Integrability and Laws of Large Numbers I

For a sequence of integrable random variables, we introduce a new set of conditions called Cesàro α-Integrability and Strong Cesàro α-Integrability and show that, for α<1/2, these conditions that are strictly weaker than Cesàro Uniform Integrability and Strong Cesàro 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, Cesàro α-integrability for appropriate α is shown to be sufficient for WLLN to hold.