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 ${\displaystyle 0\leq \theta \leq 1}$, then

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

Proof: First,

${\displaystyle \operatorname {E} [Z]=\operatorname {E} [Z\,\mathbf {1} _{\{Z\leq \theta \operatorname {E} [Z]\}}]+\operatorname {E} [Z\,\mathbf {1} _{\{Z>\theta \operatorname {E} [Z]\}}].}$

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

Related inequalities

The Paley–Zygmund inequality can be written as

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

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

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

which, after rearranging, implies that

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

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

${\displaystyle \operatorname {P} (Z>\mu -\theta \sigma )\geq {\frac {\theta ^{2}}{1+\theta ^{2}}},}$

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

References

• R. E. A. C. Paley and A. Zygmund, "On some series of functions, (3)," Proc. Camb. Phil. Soc. 28 (1932), 190–205, (cf. Lemma 19 page 192).
• 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