# Lévy inequality

An inequality for the distribution of the maximum of sums of independent random variables, centred around the corresponding medians. Let $X _ {1} \dots X _ {n}$ be independent random variables, let $S _ {k} = \sum _ {i=} 1 ^ {k} X _ {i}$ and let $mX$ be the median (cf. Median (in statistics)) of the random variable $X$; then for any $x$ one has the Lévy inequalities

$${\mathsf P} \left \{ \max _ {1\leq k \leq n } ( S _ {k} - m ( S _ {k} - S _ {n} ) ) \geq x \right \} \leq 2 {\mathsf P} \{ S _ {n} \geq x \}$$

and

$${\mathsf P} \left \{ \max _ {1\leq k \leq n } | S _ {k} - m ( S _ {k} - S _ {n} ) | \geq x \right \} \leq 2 {\mathsf P} \{ | S _ {n} | \geq x \} .$$

Immediate consequences of these inequalities are the Lévy inequalities for symmetrically-distributed random variables $X _ {1} \dots X _ {n}$:

$${\mathsf P} \left \{ \max _ {1 \leq k \leq n } S _ {k} \geq x \right \} \leq 2 {\mathsf P} \{ S _ {n} \geq x \}$$

and

$${\mathsf P} \left \{ \max _ {1 \leq k \leq n } | S _ {k} | \geq x \right \} \leq 2 {\mathsf P} \{ | S _ {n} | \geq x \} .$$

The Lévy inequalities can be regarded as generalizations of the Kolmogorov inequality. The Lévy inequalities were obtained by P. Lévy  in the investigation of general problems on the convergence of distributions of sums of independent random variables to stable laws. There is also a generalization of them to martingales .

How to Cite This Entry:
Lévy inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%A9vy_inequality&oldid=47736
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article