Hsu–Robbins–Erdős theorem explained

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 andS _{n} = X _{1} + … + X _{n,}

then

\sum\limits _{n} P( | S _{n} | > \varepsilon n) < inf ty

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.^[1] Hsu and Robbins further conjectured in ^[2] that the condition of finiteness of the variance of

X is also a necessary condition for \sum\limits _{n} P(| S _{n} | > \varepsilon n) < inf ty 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]

Notes and 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. http://samos.univ-paris1.fr/IMG/pdf_Tudor_expo-ConfMAGDA.pdf Hsu-Robbins theorem for the correlated sequences