Hsu–Robbins–Erdős theorem

In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if X 1 , … , X n is a sequence of i.i.d. random variables with zero mean and finite variance and

then

for every ε > 0 .

The result was proved by Pao-Lu Hsu and Herbert Robbins in 1947.

This is an interesting strengthening of the classical strong law of large numbers in the direction of the Borel–Cantelli lemma. The idea of such a result is probably due to Robbins, but the method of proof is vintage Hsu. Hsu and Robbins further conjectured in that the condition of finiteness of the variance of X is also a necessary condition for ∑ n ⩾ 1 P ( | S n | > ε n ) < ∞ to hold. Two years later, the famed mathematician Paul Erdős proved the conjecture.

Since then, many authors extended this result in several directions.