# 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 [1] 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 [2].

#### References

 [1] P. Lévy, "Théorie de l'addition des variables aléatoires" , Gauthier-Villars (1937) [2] M. Loève, "Probability theory" , Princeton Univ. Press (1963)
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