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]

References

1. Chung, K. L. (1979). Hsu's work in probability. The Annals of Statistics, 479–483.
2. Hsu, P. L., & Robbins, H. (1947). Complete convergence and the law of large numbers. Proceedings of the National Academy of Sciences of the United States of America, 33(2), 25.
3. Erdos, P. (1949). On a theorem of Hsu and Robbins. The Annals of Mathematical Statistics, 286–291.
4. Hsu-Robbins theorem for the correlated sequences