Chung–Erdős inequality explained

In probability theory, the Chung–Erdős inequality provides a lower bound on the probability that one out of many (possibly dependent) events occurs. The lower bound is expressed in terms of the probabilities for pairs of events.

Formally, let

A1,\ldots,An

be events. Assume that

\Pr[Ai]>0

for some

i

. Then

\Pr[A1\vee\vee

A
n] \geq
n
\left(\sum
2
\Pr[A
i]\right)
i=1
n
\sum\Pr[Ai\wedgeAj]
j=1

.

The inequality was first derived by Kai Lai Chung and Paul Erdős (in,[1] equation (4)). It was stated in the form given above by Petrov (in,[2] equation (6.10)). It can be obtained by applying the Paley–Zygmund inequality to the number of

Ai

which occur.

Notes and References

  1. Chung. K. L.. Erdös. P.. 1952-01-01. On the application of the Borel–Cantelli lemma. Transactions of the American Mathematical Society. 72. 1. 179–186. 10.1090/S0002-9947-1952-0045327-5. 0002-9947. free.
  2. Book: Petrov, Valentin Vladimirovich. Limit theorems of probability theory : sequences of independent random variables. 1995-01-01. Clarendon Press. 301554906.