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 $0\leq \theta \leq 1$, then

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

Proof: First,

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

The first addend is at most $\theta \operatorname {E} [Z]$, 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

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

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

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

which, after rearranging, implies that

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

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

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

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

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

$\operatorname {P} (Z>\theta \operatorname {E} [Z\mid Z>0])\geq {\frac {(1-\theta )^{2}\,\operatorname {E} [Z]^{2}}{\operatorname {E} [Z^{2}]}}$

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

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

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.

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. Bibcode:1932PCPS...28..190P. doi:10.1017/S0305004100010860. S2CID 178702376.
• 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. Bibcode:1932PCPS...28..266P. doi:10.1017/S0305004100010112. S2CID 122832495.