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In the [[Mathematics|mathematical]] [[Probability theory|theory of probability]], the '''Hsu–Robbins–Erdős theorem''' states that if <math>X_1, \ldots ,X_n</math> is a sequence of i.i.d. [[Random variable|random variable]]s with zero mean and finite variance and
: <math>S_n = X_1 +  \cdots  + X_n, \, </math>

then

: <math>\sum\limits_{n \geqslant 1} P( | S_n | > \varepsilon n) < \infty</math>

for every <math>\varepsilon  > 0</math>.

The result was proved by [[Biography:Pao-Lu Hsu|Pao-Lu Hsu]] and [[Biography:Herbert Robbins|Herbert Robbins]] in 1947.

This is an interesting strengthening of the classical strong [[Law of large numbers|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.<ref>Chung, K. L. (1979). Hsu's work in probability. The Annals of Statistics, 479–483.</ref> Hsu and Robbins further conjectured in <ref>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.</ref> that the condition of finiteness of the variance of <math>X</math> is also a necessary condition for <math>\sum\limits_{n \geqslant 1} P(| S_n | > \varepsilon n) < \infty</math>  to hold. Two years later, the famed mathematician [[Biography:Paul Erdős|Paul Erdős]] proved the conjecture.<ref>Erdos, P. (1949). On a theorem of Hsu and Robbins. The Annals of Mathematical Statistics, 286–291.</ref>

Since then, many authors extended this result in several directions.<ref>[http://samos.univ-paris1.fr/IMG/pdf_Tudor_expo-ConfMAGDA.pdf Hsu-Robbins theorem for the correlated sequences]</ref>

==References==
{{Reflist}}

{{DEFAULTSORT:Hsu-Robbins-Erdos Theorem}}
[[Category:Theorems in measure theory]]
[[Category:Probabilistic inequalities]]

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