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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

P ⁡ ( Z > θ E ⁡ [ Z ] ) ≥ ( 1 − θ ) 2 E ⁡ [ Z ] 2 E ⁡ [ Z 2 ] .

Proof: First,

E ⁡ [ Z ] = E ⁡ [ Z 1 { Z ≤ θ E ⁡ [ Z ] } ] + E ⁡ [ Z 1 { Z > θ E ⁡ [ Z ] } ] .

The first addend is at most θ E ⁡ [ Z ] , while the second is at most E ⁡ [ Z 2 ] 1 / 2 P ⁡ ( Z > θ E ⁡ [ Z ] ) 1 / 2 by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎

The Paley–Zygmund inequality can be written as

P ⁡ ( Z > θ E ⁡ [ Z ] ) ≥ ( 1 − θ ) 2 E ⁡ [ Z ] 2 Var ⁡ Z + E ⁡ [ Z ] 2 .

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

E ⁡ [ Z − θ E ⁡ [ Z ] ] ≤ E ⁡ [ ( Z − θ E ⁡ [ Z ] ) 1 { Z > θ E ⁡ [ Z ] } ] ≤ E ⁡ [ ( Z − θ E ⁡ [ Z ] ) 2 ] 1 / 2 P ⁡ ( Z > θ E ⁡ [ Z ] ) 1 / 2

which, after rearranging, implies that

P ⁡ ( Z > θ E ⁡ [ Z ] ) ≥ ( 1 − θ ) 2 E ⁡ [ Z ] 2 E ⁡ [ ( Z − θ E ⁡ [ Z ] ) 2 ] = ( 1 − θ ) 2 E ⁡ [ Z ] 2 Var ⁡ Z + ( 1 − θ ) 2 E ⁡ [ Z ] 2 .

This inequality is sharp; equality is achieved if Z almost surely equals a positive constant.

References

Paley–Zygmund inequality Wikipedia
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