# Sums of independent random variables with unbounded variance

**Conjecture**If are independent random variables with , then

In comparison to most probabilistic inequalities (like Hoeffding's), Feige's inequality does not deteriorate as goes to infinity, something that is useful for computer scientists.

Let . Feige argued that to prove the conjecture, one only needs to prove it for the case when and each variable has the entire probability mass distributed on 0 and for some . He proved that and conjectured that the constant 1/13 may be replaced with . It was further conjectured that "the worst case" would be one of

- \item one variable has as maximum value and the remaining random variables are always 1 (hence the probability that the sum is less than is ), \item each variable has as maximum (hence the probability that the sum is less than is ).

One way to initiate an attack on this problem is to assume and argue that the case when each variable assumes with probability and otherwise 0 is indeed the worst.

## Bibliography

*[F04] Uriel Feige: On sums of independent random variables with unbounded variance, and estimating the average degree in a graph, STOC '04: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing (2004), pp. 594 - 603. ACM

*[F05] Uriel Feige: On sums of independent random variables with unbounded variance, and estimating the average degree in a graph, Manuscript, 2005, [pdf]

The problem was also referenced at population algorithms, the blog.

* indicates original appearance(s) of problem.