MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $A_1,A_2,\dots,A_n$ be events on a probability space. For $0 \leq k \leq n$ let \begin{equation*} S_k=\sum_{1 \le {i_1}<{i_2}<\cdots<{i_k} \leq n} P(A_{i_1} \cap \cdots \cap A_{i_k}). \end{equation*} This is the $k$-th binomial moment of the number $m_n$ of those $A$s which occur.

Question: What bounds are known for $S_k$ in terms of $S_0,S_1,\dots,S_{k-1}$?

Note: Bonferroni-type inequalities give bounds for $P(m_n \geq t)$ in terms of linear combinations of the $S_i$ (for example Galambos, "Bonferroni Inequalities", Annals of Probability, Vol. 5 (1977), 577--581). It is possible to use these to give bounds for $S_k$ in terms of $S_0,S_1,\dots,S_{k-1}$ and $P(m_m \geq t)$ and use the fact that $0 \leq P(m_n \geq t) \leq 1$ to deduce bounds on $S_k$ in terms of $S_0,S_1,\dots,S_{k-1}$. It seems to me however that a lot is wasted that way. I am wondering if there are known ways of obtaining sharper bounds.

share|cite|improve this question

There are better methods using tree and graphs, for example, "Probability Bounds with Cherry Trees" in Math. of Oper. Research in 2001.

share|cite|improve this answer
Thanks! This paper looks interesting; I'll check it out. – Siksek Mar 3 '13 at 22:12

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.