Paley–Zygmund inequality

Template:Short description 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

$\mathrm{P}(Z>\theta \mathrm{E}[Z])\ge (1-\theta {)}^2\frac{\mathrm{E}[Z{]}^2}{\mathrm{E}[Z^2]}.$

Proof: First,

$\mathrm{E}[Z]=\mathrm{E}[Z{\mathrm{𝟏}}_{\{Z\le \theta \mathrm{E}[Z]\}}]+\mathrm{E}[Z{\mathrm{𝟏}}_{\{Z>\theta \mathrm{E}[Z]\}}].$

The first addend is at most $\theta \mathrm{E}[Z]$, while the second is at most $\mathrm{E}[Z^2{]}^{1/2}\mathrm{P}(Z>\theta \mathrm{E}[Z]{)}^{1/2}$ by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎

Related inequalities

The Paley–Zygmund inequality can be written as

$\mathrm{P}(Z>\theta \mathrm{E}[Z])\ge \frac{(1-\theta {)}^2\mathrm{E}[Z{]}^2}{\mathrm{Var}Z+\mathrm{E}[Z{]}^2}.$

This can be improved since, by the Cauchy–Schwarz inequality,

$\mathrm{E}[Z-\theta \mathrm{E}[Z]]\le \mathrm{E}[(Z-\theta \mathrm{E}[Z]){\mathrm{𝟏}}_{\{Z>\theta \mathrm{E}[Z]\}}]\le \mathrm{E}[(Z-\theta \mathrm{E}[Z]{)}^2{]}^{1/2}\mathrm{P}(Z>\theta \mathrm{E}[Z]{)}^{1/2}$

which, after rearranging, implies that

$\mathrm{P}(Z>\theta \mathrm{E}[Z])\ge \frac{(1-\theta {)}^2\mathrm{E}[Z{]}^2}{\mathrm{E}[(Z-\theta \mathrm{E}[Z]{)}^2]}=\frac{(1-\theta {)}^2\mathrm{E}[Z{]}^2}{\mathrm{Var}Z+(1-\theta {)}^2\mathrm{E}[Z{]}^2}.$

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

$\mathrm{P}(Z>\mu -\theta \sigma )\ge \frac{{\theta }^2}{1+{\theta }^2},$

where $\mu =\mathrm{E}[Z]$ and ${\sigma }^2=\mathrm{Var}[Z]$. This follows from the substitution $\theta =1-{\theta }^{\prime }\sigma /\mu$ 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

$\mathrm{P}(Z>\theta \mathrm{E}[Z\mid Z>0])\ge \frac{(1-\theta {)}^2\mathrm{E}[Z{]}^2}{\mathrm{E}[Z^2]}$

for every $0\le \theta \le 1$. 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 $\mathrm{P}(Z>0)$ 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

$\mathrm{P}(Z>\theta \mathrm{E}[Z\mid Z>0])\ge \frac{(1-\theta {)}^{p/(p-1)}\mathrm{E}[Z{]}^{p/(p-1)}}{\mathrm{E}[Z^p{]}^{1/(p-1)}}.$

for every $0\le \theta \le 1$. 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.

Further reading

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