Application of Convergence Theorems

We present some applications of the results from Chap. 5. We show how Kolmogorov 0-1 law and Hewitt-Savage 0-1 law follow from the reverse martingale convergence theorem. The strong law of large numbers for average of independent and identically distributed (iid) random variables, as well as for U-statistics, are proved by using reverse martingales. The strong law for exchangeable sequences is also established. We also state and prove de-Finetti’s theorem for exchangeable random variables, which says that every such sequence is iid, conditional on an appropriate \(\sigma \) -field. The final topic in this chapter is Kakutani’s theorem for product martingales, which has application in statistics.