Hsu–Robbins–Erdős theorem
In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if [math]\displaystyle{ X_1, \ldots ,X_n }[/math] is a sequence of i.i.d. random variables with zero mean and finite variance and
- [math]\displaystyle{ S_n = X_1 + \cdots + X_n, \, }[/math]
then
- [math]\displaystyle{ \sum\limits_{n \geqslant 1} P( | S_n | \gt \varepsilon n) \lt \infty }[/math]
for every [math]\displaystyle{ \varepsilon \gt 0 }[/math].
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 [math]\displaystyle{ X }[/math] is also a necessary condition for [math]\displaystyle{ \sum\limits_{n \geqslant 1} P(| S_n | \gt \varepsilon n) \lt \infty }[/math] 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
- ↑ Chung, K. L. (1979). Hsu's work in probability. The Annals of Statistics, 479–483.
- ↑ 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.
- ↑ Erdos, P. (1949). On a theorem of Hsu and Robbins. The Annals of Mathematical Statistics, 286–291.
- ↑ Hsu-Robbins theorem for the correlated sequences
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