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Hsu–Robbins–Erdős theorem

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Statement in probability theory

In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if X 1 , … , X n {\displaystyle 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 + ⋯ + X n , {\displaystyle S_{n}=X_{1}+\cdots +X_{n},\,}

then

∑ n ⩾ 1 P ( | S n | > ε n ) < ∞ {\displaystyle \sum \limits _{n\geqslant 1}P(|S_{n}|>\varepsilon n)<\infty }

for every ε > 0 {\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. Hsu and Robbins further conjectured in that the condition of finiteness of the variance of X {\displaystyle X} is also a necessary condition for ∑ n ⩾ 1 P ( | S n | > ε n ) < ∞ {\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.

Since then, many authors extended this result in several directions.

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
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