We provide a relatively simple proof of the generalization of the Hsu-Robbins-Erdős theorem by Baum and Katz (1965). Our proof avoids using some advanced inequalities, and is essentially based on the idea used to prove the strong law of large numbers under finite fourth moment condition.
The strong law of large numbers heavily depends on the fact that each of appears repeatedly in the sequence . We prove a version of the strong law of large number in which no random variable gets “recycled.” More precisely, we prove the Hsu-Robbins-Erdős theorem, which states that for i.i.d. , we have if andContinue reading “The Strong Law of Large Numbers without Recycling”
We simplify our probabilistic proof of a Wallis-type formula for the Gamma function by using Gamma distributions instead of normal distributions.