# 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=55237