In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its first two moments. The inequality was proved by Raymond Paley and Antoni Zygmund.

Theorem: If Z  0 is a random variable with finite variance, and if , then

Proof: First,

The first addend is at most , while the second is at most by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎

The Paley–Zygmund inequality can be written as

This can be improved. By the Cauchy–Schwarz inequality,

which, after rearranging, implies that


This inequality is sharp; equality is achieved if Z almost surely equals a positive constant.

In turn, this implies another convenient form (known as Cantelli's inequality) which is

where and . This follows from the substitution valid when .

A strengthened form of the Paley-Zygmund inequality states that if Z is a non-negative random variable then

for every . This inequality follows by applying the usual Paley-Zygmund inequality to the conditional distribution of Z given that it is positive and noting that the various factors of cancel.

Both this inequality and the usual Paley-Zygmund inequality also admit versions:[1] If Z is a non-negative random variable and then

for every . This follows by the same proof as above but using Hölder's inequality in place of the Cauchy-Schwarz inequality.

See also

References

  1. Petrov, Valentin V. (1 August 2007). "On lower bounds for tail probabilities". Journal of Statistical Planning and Inference. 137 (8): 2703–2705. doi:10.1016/j.jspi.2006.02.015.

Further reading


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