Paley–Zygmund inequality

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In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its mean and variance (i.e., 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 < θ < 1, then


\operatorname{P}( Z \ge \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 < \theta \operatorname{E}[Z] \}}]  + \operatorname{E}[ Z \, \mathbf{1}_{\{ Z \ge \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 \ge \theta\operatorname{E}[Z])^{1/2} by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎

Related inequalities[edit]

The Paley–Zygmund inequality can be written as


\operatorname{P}( Z \ge \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 \ge \theta \operatorname{E}[Z] \}} ]
\le \operatorname{E}[ (Z - \theta \operatorname{E}[Z])^2 ]^{1/2} \operatorname{P}( Z \ge \theta \operatorname{E}[Z] )^{1/2}

which, after rearranging, implies that


\operatorname{P}(Z \ge \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, for example.

References[edit]

  • R.E.A.C.Paley and A.Zygmund, A note on analytic functions in the unit circle, Proc. Camb. Phil. Soc. 28, 1932, 266–272