Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables

This paper provides L¹ and weak laws of large numbers for uniformly integrable L¹-mixingales. The L¹-mixingale condition is a condition of asymptotic weak temporal dependence that is weaker than most conditions considered in the literature. Processes covered by the laws of large numbers include martingale difference, ø(·), ρ(·), and α(·) mixing, autoregressive moving average, infinite-order moving average, near epoch dependent, L¹-near epoch dependent, and mixingale sequences and triangular arrays. The random variables need not possess more than one finite moment and the L¹-mixingale numbers need not decay to zero at any particular rate. The proof of the results is remarkably simple and completely self-contained.