Hsu–Robbins–Erdős theorem

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

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

then

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

for every ${\displaystyle \varepsilon >0}$.

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.^[1] Hsu and Robbins further conjectured in ^[2] that the condition of finiteness of the variance of ${\displaystyle X}$ is also a necessary condition for ${\displaystyle \sum \limits _{n\geqslant 1}P(|S_{n}|>\varepsilon n)<\infty }$ to hold. Two years later, the famed mathematician Paul Erdős proved the conjecture.^[3]

Since then, many authors extended this result in several directions.^[4]