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. ∎

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 ${\displaystyle 0\leq \mu -\theta \sigma \leq \mu }$.

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

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

for every ${\displaystyle 0\leq \theta \leq 1}$. 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 ${\displaystyle \operatorname {P} (Z>0)}$ cancel.

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

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

for every ${\displaystyle 0\leq \theta \leq 1}$. This follows by the same proof as above but using Hölder's inequality in place of the Cauchy-Schwarz inequality.

• Cantelli's inequality
• Second moment method
• Concentration inequality – a summary of tail-bounds on random variables.

• 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.