Hsu–Robbins–Erdős theorem

In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if $X_{1},\ldots ,X_{n}$ is a sequence of i.i.d. random variables with zero mean and finite variance and

$S_{n}=X_{1}+\cdots +X_{n},\,$

then

$\sum \limits _{n\geqslant 1}P(|S_{n}|>\varepsilon n)<\infty$

for every $\varepsilon >0$.

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 $\sum \limits _{n\geqslant 1}P(|S_{n}|>\varepsilon n)<\infty$ to hold. Two years later, the famed mathematician Paul Erdős proved the conjecture.

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