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 [math]\displaystyle{ 0 \le \theta \le 1 }[/math], then

[math]\displaystyle{ \operatorname{P}( Z \gt \theta\operatorname{E}[Z] ) \ge (1-\theta)^2 \frac{\operatorname{E}[Z]^2}{\operatorname{E}[Z^2]}. }[/math]

Proof: First,

[math]\displaystyle{ \operatorname{E}[Z] = \operatorname{E}[ Z \, \mathbf{1}_{\{ Z \le \theta \operatorname{E}[Z] \}}] + \operatorname{E}[ Z \, \mathbf{1}_{\{ Z \gt \theta \operatorname{E}[Z] \}} ]. }[/math]

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

Related inequalities

The Paley–Zygmund inequality can be written as

[math]\displaystyle{ \operatorname{P}( Z \gt \theta \operatorname{E}[Z] ) \ge \frac{(1-\theta)^2 \, \operatorname{E}[Z]^2}{\operatorname{Var} Z + \operatorname{E}[Z]^2}. }[/math]

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

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

which, after rearranging, implies that

[math]\displaystyle{ \operatorname{P}(Z \gt \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}. }[/math]

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

[math]\displaystyle{ \operatorname{P}(Z \gt \mu - \theta \sigma) \ge \frac{\theta^2}{1+\theta^2}, }[/math]

where [math]\displaystyle{ \mu=\operatorname{E}[Z] }[/math] and [math]\displaystyle{ \sigma^2 = \operatorname{Var}[Z] }[/math]. This follows from the substitution [math]\displaystyle{ \theta = 1-\theta'\sigma/\mu }[/math] valid when [math]\displaystyle{ 0\le \mu - \theta \sigma\le\mu }[/math].

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

[math]\displaystyle{ \operatorname{P}( Z \gt \theta \operatorname{E}[Z \mid Z \gt 0] ) \ge \frac{(1-\theta)^2 \, \operatorname{E}[Z]^2}{\operatorname{E}[Z^2]} }[/math]

for every [math]\displaystyle{ 0 \leq \theta \leq 1 }[/math]. 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 [math]\displaystyle{ \operatorname{P}(Z\gt 0) }[/math] cancel.

Both this inequality and the usual Paley-Zygmund inequality also admit [math]\displaystyle{ L^p }[/math] versions: If Z is a non-negative random variable and [math]\displaystyle{ p \gt 1 }[/math] then

[math]\displaystyle{ \operatorname{P}( Z \gt \theta \operatorname{E}[Z \mid Z \gt 0] ) \ge \frac{(1-\theta)^{p/(p-1)} \, \operatorname{E}[Z]^{p/(p-1)}}{\operatorname{E}[Z^p]^{1/(p-1)}}. }[/math]

for every [math]\displaystyle{ 0 \leq \theta \leq 1 }[/math]. 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

• Paley, R. E. A. C.; Zygmund, A. (April 1932). "On some series of functions, (3)". Mathematical Proceedings of the Cambridge Philosophical Society 28 (2): 190–205. doi:10.1017/S0305004100010860. Bibcode: 1932PCPS...28..190P. 
• Paley, R. E. A. C.; Zygmund, A. (July 1932). "A note on analytic functions in the unit circle". Mathematical Proceedings of the Cambridge Philosophical Society 28 (3): 266–272. doi:10.1017/S0305004100010112. Bibcode: 1932PCPS...28..266P.