Paley–Zygmund inequality
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 which is
where and.
This follows from the substitution valid when.
