Hsu–Robbins–Erdős Theorem

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 and

: $S_n = X_1 + \cdots + X_n, \,$

then

: $\sum\limits_ P( , S_n , > \varepsilon n) < \infty$

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. Hsu and Robbins further conjectured in that the condition of finiteness of the variance of $X$ is also a necessary condition for $\sum\limits_ P(, S_n , > \varepsilon n) < \infty$ to hold. Two years later, the famed mathematician Paul Erdős proved the conjecture.

Since then, many authors extended this result in several directions.Hsu-Robbins theorem for the correlated sequences
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References

{{DEFAULTSORT:Hsu-Robbins-Erdos Theorem
Theorems in measure theory
Probabilistic inequalities