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 $0 \le \theta \le 1$, then

:
\operatorname( Z > \theta\operatorname )
\ge (1-\theta)^2 \frac.

Proof: First,
:
\operatorname = \operatorname Z \, \mathbf_ + \operatorname Z \, \mathbf_

The first addend is at most \theta \operatorname /math>, while the second is at most \operatorname ^2 \operatorname( Z > \theta\operatorname ^ by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎

Related inequalities

The Paley–Zygmund inequality can be written as

:
\operatorname( Z > \theta \operatorname )
\ge \frac.

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

:
\operatorname - \theta \operatorname[Z
\le \operatorname[ (Z - \theta \operatorname \mathbf_ ">.html" ;"title=" - \theta \operatorname[Z"> - \theta \operatorname[Z
\le \operatorname[ (Z - \theta \operatorname \mathbf_ \le \operatorname[ (Z - \theta \operatorname ^2 ]^ \operatorname( Z > \theta \operatorname )^

which, after rearranging, implies that

:
\operatorname(Z > \theta \operatorname
\ge \frac
= \frac.

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
:$\operatorname(Z > \mu - \theta \sigma) \ge \frac,$
where \mu=\operatorname /math> and \sigma^2 = \operatorname /math>.
This follows from the substitution $\theta = 1-\theta'\sigma/\mu$ valid when $0\le \mu - \theta \sigma\le\mu$.

A strengthened form of the Paley-Zygmund inequality states that if Z is a non-negative random variable then
:
\operatorname( Z > \theta \operatorname \mid Z > 0)
\ge \frac

for every $0 \leq \theta \leq 1$.
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 $\operatorname(Z>0)$ cancel.

Both this inequality and the usual Paley-Zygmund inequality also admit $L^p$ versions: If Z is a non-negative random variable and $p > 1$ then
:
\operatorname( Z > \theta \operatorname \mid Z > 0)
\ge \frac.

for every $0 \leq \theta \leq 1$. This follows by the same proof as above but using Hölder's inequality in place of the Cauchy-Schwarz inequality.

See also

* Cantelli's inequality
* Second moment method
* Concentration inequality – a summary of tail-bounds on random variables.

References

Further reading

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{{DEFAULTSORT:Paley-Zygmund inequality
Probabilistic inequalities