Paley–Zygmund inequality

In mathematics, the Paley - Zygmund inequality bounds theprobability that a positive random variable is small, in terms ofits mean and variance (i.e., its first two moments). The inequality wasproved by Raymond Paley and Antoni Zygmund.

Theorem: If "Z" &ge; 0 is a random variable withfinite variance, and if 0 < &theta; < 1, then

:$Pr\; lbrace\; Z\; geq\; heta,\; operatorname\{E\}(Z)\; brace\; geq\; (1-\; heta)^2,\; frac\{(operatorname\{E\}(Z))^2\}\{operatorname\{E\}(Z^2)\}.$

Proof: First, :Obviously, the first addend is at most $heta\; operatorname\{E\}(Z)$. The second one is at most:$lbrace\; operatorname\{E\}\; Z^2\; brace^\{1/2\}\; lbrace\; operatorname\{E\}\; mathbf\{1\}\_\{Z\; geq\; heta\; operatorname\{E\}\; Z\}\; brace^\{1/2\}\; =\; Big(\; operatorname\{E\}\; Z^2\; Big)^\{1/2\}\; Big(Pr\; lbrace\; Z\; geq\; heta,\; operatorname\{E\}(Z)\; braceBig)^\{1/2\}$according to the Cauchy-Schwarz inequality. ∎

Related inequalities

The right-hand side of the Paley - Zygmund inequality can be written as:$Pr\; lbrace\; Z\; geq\; heta,\; operatorname\{E\}(Z)\; brace\; geq\; frac\{(1-\; heta)^2\; ,\; (operatorname\{E\}(Z))^2\}\{(operatorname\{E\}(Z))^2\; +\; operatorname\{Var\}\; Z\}.$

The one-sided Chebyshev inequality gives a slightly better bound::$Pr\; lbrace\; Z\; geq\; heta,\; operatorname\{E\}(Z)\; brace\; geq\; frac\{(1-\; heta)^2\; ,\; (operatorname\{E\}(Z))^2\}\{(1-\; heta)^2\; ,\; (operatorname\{E\}(Z))^2+\; operatorname\{Var\}\; Z\}.$The latter is sharp.

References

* 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