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

Proof: First,

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

Related inequalities
The Paley–Zygmund inequality can be written as



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

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



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

which, after rearranging, implies that



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

\operatorname{P}(Z > \mu - \theta \sigma) \ge \frac{\theta^2}{1+\theta^2}, $$ where $$\mu=\operatorname{E}[Z]$$ and $$\sigma^2 = \operatorname{Var}[Z]$$. This follows from the substitution $$\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{P}( Z > \theta \operatorname{E}[Z \mid Z > 0] ) \ge \frac{(1-\theta)^2 \, \operatorname{E}[Z]^2}{\operatorname{E}[Z^2]} $$ for every $$ 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{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

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