The weak and strong laws of large numbers

converges in probability to 0. A strong law of large numbers is a statement that (1) converges almost surely to 0. Thus, if the hypotheses assumed on the sequence of random variables are the same, a strong law implies a weak law. We shall prove the weak law of large numbers for a sequence of independent identically distributed L random variables, and the strong law of large for the same hypotheses. We give separate proofs for these theorems as an occasion to inspect different machinery, although to establish the weak law it thus suffices to prove the strong law. One reason to distinguish these laws is for cases when we impose different hypotheses. We also prove Markov’s weak law of large numbers, which states that if Xn is a sequence of L 2 random variables that are pairwise uncorrelated and